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pavages_4xn.html
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pavages_4xn.html
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<h1 class="mume-header undefined" id="pavages-dun-rectangle-4n-par-des-rectangles-daire-4">Pavages d'un rectangle 4×n par des rectangles d'aire 4</h1>
<p>Auteur : Franck CHAMBON</p>
<div class="code-chunk" data-id="pa" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks2b48269e-32ec-47df-af19-ed7c753b6a5a td {border: 1px solid white;}</style><table id="blocks2b48269e-32ec-47df-af19-ed7c753b6a5a" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
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Color: (175, 100, 100)" style="width: 50px; height: 50px;background-color: rgb(175, 100, 100);"></td><td title="Index: [1, 4]
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Color: (0, 0, 139)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 139);"></td><td title="Index: [3, 1]
Color: (0, 128, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 128, 0);"></td><td title="Index: [3, 2]
Color: (0, 128, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 128, 0);"></td><td title="Index: [3, 3]
Color: (0, 128, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 128, 0);"></td><td title="Index: [3, 4]
Color: (0, 128, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 128, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 255)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 255);"></td><td title="Index: [3, 6]
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</div></div><blockquote>
<p>Exemple de pavage d'un rectangle <span class="mathjax-exps">$4\times 8$</span> par des rectangles d'aire <span class="mathjax-exps">$4$</span>.</p>
<blockquote>
<p><strong>Combien y a-t-il de tels pavages ?</strong><br>
On notera <span class="mathjax-exps">$a_n$</span> le nombre de pavages pour un rectangle <span class="mathjax-exps">$4\times n$</span>.</p>
</blockquote>
</blockquote>
<div class="code-chunk" data-id="code-chunk-id-1" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks6d838d27-3c6a-444e-bb8f-d4e14009bfba td {border: 1px solid white;}</style><table id="blocks6d838d27-3c6a-444e-bb8f-d4e14009bfba" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
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Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [2, 5]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [2, 6]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [2, 7]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [2, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [2, 9]
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Color: (0, 0, 255)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 255);"></td><td title="Index: [3, 1]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 2]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 3]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 4]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 5]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 6]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 7]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 8]
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</div></div><blockquote>
<p>Les trois motifs : les rectangles d'aire 4, à côtés entiers.</p>
</blockquote>
<hr>
<p>Ce problème est résolu de manière progressive, et c'est l'occasion d'utiliser :</p>
<ul>
<li>des méthodes de dénombrements de pavages ;</li>
<li>des suites récurrentes linéaires (autres que la très classique suite de Fibonacci) ;</li>
<li>des algorithmes de calcul de termes d'une telle suite :
<ul>
<li>par calcul successif ;</li>
<li>par calcul matriciel ;</li>
<li>par calcul polynomial.</li>
</ul>
</li>
</ul>
<blockquote>
<p>Table des matières</p>
</blockquote>
<ul>
<li><a href="#probl%C3%A8me-simplifi%C3%A9">Problème simplifié</a></li>
<li><a href="#cas-g%C3%A9n%C3%A9ral">Cas général</a>
<ul>
<li><a href="#premi%C3%A8res-valeurs">Premières valeurs</a></li>
<li><a href="#r%C3%A9currence-pour-a_n">Récurrence pour <span class="mathjax-exps">$a_n$</span></a></li>
<li><a href="#r%C3%A9currence-pour-b_n">Récurrence pour <span class="mathjax-exps">$b_n$</span></a></li>
<li><a href="#r%C3%A9currence-pour-c_n">Récurrence pour <span class="mathjax-exps">$c_n$</span></a></li>
</ul>
</li>
<li><a href="#calcul-des-valeurs-suivantes">Calcul des valeurs suivantes</a>
<ul>
<li><a href="#relation-dordre-8-pour-a_n">Relation d'ordre <span class="mathjax-exps">$8$</span> pour <span class="mathjax-exps">$a_n$</span></a>
<ul>
<li><a href="#dapr%C3%A8s-oeissupsup">D'après OEIS®</a></li>
<li><a href="#preuve-de-cette-relation-de-r%C3%A9currence">Preuve de cette relation de récurrence</a></li>
</ul>
</li>
<li><a href="#calculs-successifs-des-premiers-termes">Calculs successifs des premiers termes</a></li>
<li><a href="#calcul-matriciel-dun-terme-lointain">Calcul matriciel d'un terme lointain</a></li>
<li><a href="#calcul-polynomial-dun-terme-lointain">Calcul polynomial d'un terme lointain</a></li>
</ul>
</li>
<li><a href="#quelques-probl%C3%A8mes-similaires">Quelques problèmes similaires</a></li>
<li><a href="#annexe-fichier-outilspy">ANNEXE : fichier <code>outils.py</code></a></li>
</ul>
<hr>
<h2 class="mume-header" id="probl%C3%A8me-simplifi%C3%A9">Problème simplifié</h2>
<ul>
<li>Si on ne pave qu'avec des barres verticales, il n'y a qu'une seule façon de paver un rectangle <span class="mathjax-exps">$4\times n$</span>.
<ul>
<li><span class="mathjax-exps">$\forall n \in \mathbb N \quad a_n = 1$</span></li>
</ul>
</li>
<li>Si on ne pave qu'avec des barres verticales et des carrés, il est facile de voir que <span class="mathjax-exps">$(a_n)_n$</span> correspond à la suite de Fibonacci décalée, à savoir :
<ul>
<li><span class="mathjax-exps">$a_0 = a_1 = 1$</span>, et <span class="mathjax-exps">$\forall n \in \mathbb N \quad a_{n+2} = a_{n+1} + a_n$</span></li>
</ul>
</li>
</ul>
<hr>
<h2 class="mume-header" id="cas-g%C3%A9n%C3%A9ral">Cas général</h2>
<p>Pour tout <span class="mathjax-exps">$n$</span> entier, on note :</p>
<ul>
<li><span class="mathjax-exps">$a_n$</span> le nombre de pavages d'un rectangle <span class="mathjax-exps">$4\times n$</span>.</li>
<li><span class="mathjax-exps">$b_n$</span> le nombre de pavages d'un rectangle <span class="mathjax-exps">$4\times n$</span> modifié :
<ul>
<li>on y ajoute un carré <span class="mathjax-exps">$2\times2$</span> à droite (en haut ou en bas).</li>
</ul>
</li>
<li><span class="mathjax-exps">$c_n$</span> le nombre de pavages d'un rectangle <span class="mathjax-exps">$4\times n$</span> modifié :
<ul>
<li>on y ajoute un carré <span class="mathjax-exps">$2\times2$</span> à droite au milieu.</li>
</ul>
</li>
</ul>
<hr>
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Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [2, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Un rectangle <span class="mathjax-exps">$4 \times 8$</span>.</p>
<blockquote>
<p>Il y a <span class="mathjax-exps">$a_8$</span> façons de le paver.</p>
</blockquote>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-3" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks0d665048-8797-46e4-8bcf-276c98557e8c td {border: 1px solid white;}</style><table id="blocks0d665048-8797-46e4-8bcf-276c98557e8c" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
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Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [0, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [1, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 8]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 9]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 8]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 9]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Un rectangle <span class="mathjax-exps">$4\times 8$</span> avec un carré <span class="mathjax-exps">$2\times2$</span> à droite en bas.</p>
<blockquote>
<p>Il y a <span class="mathjax-exps">$b_8$</span> façons de le paver.</p>
</blockquote>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-4" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks7db2fd1d-2bdb-40a7-b348-18f758da8d34 td {border: 1px solid white;}</style><table id="blocks7db2fd1d-2bdb-40a7-b348-18f758da8d34" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [0, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 8]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 9]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 8]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 9]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Un rectangle <span class="mathjax-exps">$4\times 8$</span> avec un carré <span class="mathjax-exps">$2\times2$</span> à droite au milieu.</p>
<blockquote>
<p>Il y a <span class="mathjax-exps">$c_8$</span> façons de le paver.</p>
</blockquote>
</blockquote>
<hr>
<p>Typiquement, ce genre de problème possède une solution avec une suite récurrente linéaire.</p>
<ul>
<li>Il est nécessaire de déterminer les premiers termes,</li>
<li>puis d'établir une formule de récurrence pour <span class="mathjax-exps">$a_n$</span>, <span class="mathjax-exps">$b_n$</span> et <span class="mathjax-exps">$c_n$</span>.</li>
</ul>
<h3 class="mume-header" id="premi%C3%A8res-valeurs">Premières valeurs</h3>
<ol>
<li>
<p><span class="mathjax-exps">$a_n$</span> pour <span class="mathjax-exps">$n<4$</span></p>
<ul>
<li><span class="mathjax-exps">$a_0 = 1$</span>, il y a une unique façon de paver le vide ;</li>
<li><span class="mathjax-exps">$a_1 = 1$</span>, une barre verticale ;</li>
<li><span class="mathjax-exps">$a_2 = 2$</span>, deux barres verticales, ou bien deux carrés ;</li>
<li><span class="mathjax-exps">$a_3 = 3$</span>, trois barres verticales, ou bien deux carrés flanqués d'une barre verticale (à gauche ou à droite).</li>
</ul>
</li>
<li>
<p><span class="mathjax-exps">$b_n$</span> pour <span class="mathjax-exps">$n<2$</span></p>
<ul>
<li><span class="mathjax-exps">$b_0 = 1$</span>, un carré ;</li>
<li><span class="mathjax-exps">$b_1 = 1$</span>, un carré et une barre verticale ;</li>
</ul>
</li>
<li>
<p><span class="mathjax-exps">$c_n$</span> pour <span class="mathjax-exps">$n<4$</span><br>
Pour <span class="mathjax-exps">$n<4$</span>, on ne peut rien mettre d'autre qu'un carré à droite, et alors on a <span class="mathjax-exps">$c_n=a_n$</span> pour <span class="mathjax-exps">$n<4$</span>.</p>
<ul>
<li><span class="mathjax-exps">$c_0 = 1$</span> ;</li>
<li><span class="mathjax-exps">$c_1 = 1$</span> ;</li>
<li><span class="mathjax-exps">$c_2 = 2$</span> ;</li>
<li><span class="mathjax-exps">$c_3 = 3$</span> ;</li>
</ul>
</li>
</ol>
<h3 class="mume-header" id="r%C3%A9currence-pour-a_n">Récurrence pour <span class="mathjax-exps">$a_n$</span></h3>
<p>Il y a plusieurs cas de motifs pour la partie droite d'un pavage de rectangle <span class="mathjax-exps">$4\times n$</span>, avec <span class="mathjax-exps">$n\geqslant 4$</span> :</p>
<ol>
<li>il y a une barre verticale ;</li>
<li>il y a <span class="mathjax-exps">$4$</span> barres horizontales ;</li>
<li>il y a <span class="mathjax-exps">$2$</span> carrés ;</li>
<li>il y a un carré et deux barres horizontales
<ol>
<li>carré en haut ;</li>
<li>carré au milieu ;</li>
<li>carré en bas.</li>
</ol>
</li>
</ol>
<p>Graphiquement, cela donne :</p>
<hr>
<div class="code-chunk" data-id="code-chunk-id-5" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks080449c1-c24f-41fd-8f6b-f0365b24b9c1 td {border: 1px solid white;}</style><table id="blocks080449c1-c24f-41fd-8f6b-f0365b24b9c1" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
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Color: (0, 0, 255)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 255);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 7]
Color: (0, 0, 255)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 255);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 7]
Color: (0, 0, 255)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 255);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 7]
Color: (0, 0, 255)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 255);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$a_{n-1}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-6" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks8b23ae88-a2c3-4b83-abdf-b3f5181d5b4d td {border: 1px solid white;}</style><table id="blocks8b23ae88-a2c3-4b83-abdf-b3f5181d5b4d" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 5]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 6]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 7]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [1, 5]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [1, 6]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [1, 7]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 5]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 6]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 7]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 5]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 6]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 7]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$a_{n-4}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-7" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocksae458a20-c23c-44a5-a2bb-7c50433e5a21 td {border: 1px solid white;}</style><table id="blocksae458a20-c23c-44a5-a2bb-7c50433e5a21" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [0, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [1, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (175, 100, 100)" style="width: 50px; height: 50px;background-color: rgb(175, 100, 100);"></td><td title="Index: [2, 7]
Color: (175, 100, 100)" style="width: 50px; height: 50px;background-color: rgb(175, 100, 100);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (175, 100, 100)" style="width: 50px; height: 50px;background-color: rgb(175, 100, 100);"></td><td title="Index: [3, 7]
Color: (175, 100, 100)" style="width: 50px; height: 50px;background-color: rgb(175, 100, 100);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$a_{n-2}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-8" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocksabff403d-a3bc-4242-8c0e-bdf138f2cdb2 td {border: 1px solid white;}</style><table id="blocksabff403d-a3bc-4242-8c0e-bdf138f2cdb2" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [0, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [1, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 5]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 6]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 7]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 5]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 6]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 7]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, <em>par symétrie horizontale</em>, il y a <span class="mathjax-exps">$b_{n-4}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-9" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocksf4916848-edfd-4ed6-b9a1-aa8fc86b55b1 td {border: 1px solid white;}</style><table id="blocksf4916848-edfd-4ed6-b9a1-aa8fc86b55b1" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 5]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 6]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 7]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [1, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [2, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 5]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 6]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 7]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$c_{n-4}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-10" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocksb1677fa2-dfac-4af1-852b-b9e2efe0facc td {border: 1px solid white;}</style><table id="blocksb1677fa2-dfac-4af1-852b-b9e2efe0facc" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 5]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 6]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [0, 7]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [1, 5]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [1, 6]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [1, 7]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [2, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [3, 7]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$b_{n-4}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<p><strong>Bilan</strong> : pour <span class="mathjax-exps">$n\geqslant 4$</span>, on a <span class="mathjax-exps">$a_n = a_{n-1} + a_{n-4} + a_{n-2} + 2b_{n-4} + c_{n-4}$</span></p>
<hr>
<h3 class="mume-header" id="r%C3%A9currence-pour-b_n">Récurrence pour <span class="mathjax-exps">$b_n$</span></h3>
<p>Il y a plusieurs cas de motifs pour la partie droite d'un pavage de rectangle <span class="mathjax-exps">$4\times n$</span> flanqué d'un carré à droite en bas, avec <span class="mathjax-exps">$n\geqslant 2$</span> :</p>
<ol>
<li>il y a un carré ;</li>
<li>il y a <span class="mathjax-exps">$2$</span> barres horizontales.</li>
</ol>
<hr>
<div class="code-chunk" data-id="code-chunk-id-11" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks80e1833c-effc-4555-a3ae-a4db4f4ed619 td {border: 1px solid white;}</style><table id="blocks80e1833c-effc-4555-a3ae-a4db4f4ed619" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [0, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [1, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 8]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [2, 9]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 8]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [3, 9]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$a_{n}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-12" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocksce3a4195-37b5-45a6-8a5e-58650c61022e td {border: 1px solid white;}</style><table id="blocksce3a4195-37b5-45a6-8a5e-58650c61022e" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [0, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [1, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 7]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 8]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [2, 9]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 7]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 8]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [3, 9]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$b_{n-2}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<p><strong>Bilan</strong> : pour <span class="mathjax-exps">$n\geqslant 2$</span>, on a <span class="mathjax-exps">$b_n = b_{n-2} + a_{n}$</span></p>
<hr>
<h3 class="mume-header" id="r%C3%A9currence-pour-c_n">Récurrence pour <span class="mathjax-exps">$c_n$</span></h3>
<p>Il y a plusieurs cas de motifs pour la partie droite d'un pavage de rectangle <span class="mathjax-exps">$4\times n$</span> flanqué d'un carré à droite au milieu, avec <span class="mathjax-exps">$n\geqslant 4$</span> :</p>
<ol>
<li>il y a un carré ;</li>
<li>il y a <span class="mathjax-exps">$2$</span> barres horizontales.</li>
</ol>
<hr>
<div class="code-chunk" data-id="code-chunk-id-13" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocks9aaa5c85-c5ff-444c-be50-842f3b3962e4 td {border: 1px solid white;}</style><table id="blocks9aaa5c85-c5ff-444c-be50-842f3b3962e4" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [0, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [0, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr><tr><td title="Index: [1, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [1, 8]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [1, 9]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [2, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 8]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td><td title="Index: [2, 9]
Color: (255, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(255, 0, 0);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 6]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 7]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$a_{n}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<div class="code-chunk" data-id="code-chunk-id-14" data-cmd="/home/francky/anaconda3/bin/python3"><div class="input-div"><div class="btn-group"><div class="run-btn btn"><span>▶︎</span></div><div class="run-all-btn btn">all</div></div><div class="status">running...</div></div><div class="output-div"><style type="text/css">table.blockgrid {border: none;} .blockgrid tr {border: none;} .blockgrid td {padding: 0px;} #blocksa1781977-e87a-49a9-bafb-0f257f741338 td {border: 1px solid white;}</style><table id="blocksa1781977-e87a-49a9-bafb-0f257f741338" class="blockgrid"><tbody><tr><td title="Index: [0, 0]
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Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 5]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [2, 6]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [2, 7]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [2, 8]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td><td title="Index: [2, 9]
Color: (25, 205, 25)" style="width: 50px; height: 50px;background-color: rgb(25, 205, 25);"></td></tr><tr><td title="Index: [3, 0]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 1]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 2]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 3]
Color: (0, 0, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 0, 0);"></td><td title="Index: [3, 4]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [3, 5]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [3, 6]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [3, 7]
Color: (0, 255, 0)" style="width: 50px; height: 50px;background-color: rgb(0, 255, 0);"></td><td title="Index: [3, 8]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td><td title="Index: [3, 9]
Color: (255, 255, 255)" style="width: 50px; height: 50px;background-color: rgb(255, 255, 255);"></td></tr></tbody></table>
</div></div><blockquote>
<p>Dans ce cas, il y a <span class="mathjax-exps">$c_{n-4}$</span> façons de paver ce qu'il reste.</p>
</blockquote>
<hr>
<p><strong>Bilan</strong> : pour <span class="mathjax-exps">$n\geqslant 4$</span>, on a <span class="mathjax-exps">$c_n = c_{n-4} + a_{n}$</span></p>
<hr>
<h2 class="mume-header" id="calcul-des-valeurs-suivantes">Calcul des valeurs suivantes</h2>
<p>On utilise les premières valeurs et les relations de récurrence :</p>
<ul>
<li>d'abord pour calculer <span class="mathjax-exps">$b_2$</span> et <span class="mathjax-exps">$b_3$</span>
<ul>
<li><span class="mathjax-exps">$b_2 = b_0 + a_2 = 1+2 = 3$</span></li>
<li><span class="mathjax-exps">$b_3 = b_1 + a_3 = 1+3 = 4$</span></li>
</ul>
</li>
<li>ensuite pour calculer des valeurs suivantes.</li>
</ul>
<div class="code-chunk" data-id="valeurs" data-cmd="python3"><div class="input-div"><pre data-role="codeBlock" data-info="python {code_chunk_offset=15, cmd="python3" run_on_save id="valeurs" output="none"}" class="language-python">a <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">]</span>
b <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">4</span><span class="token punctuation">]</span>
c <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">]</span>