与方块种类有关的数列

葡萄糖 · 发表于 2019-3-23 19:43:39

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本帖最后由葡萄糖于 2019-3-23 21:19 编辑

设数列\(a_{n}\)表示\(n\)个小正方形能拼成互不全等图形的个数
\begin{align*}
&n=1&&\blacksquare\\
&n=2&&\blacksquare\blacksquare\\
&n=3&&\blacksquare\blacksquare\blacksquare\\
&&&\overset{\Large{\blacksquare}}{\blacksquare}\blacksquare\\
&n=4&&\blacksquare\blacksquare\blacksquare\blacksquare\\
&&&\overset{\Large{\blacksquare}}{\blacksquare}\blacksquare\blacksquare&&\blacksquare\overset{\Large{\blacksquare}}{\blacksquare}\blacksquare\\
&&&\blacksquare\overset{\Large{\blacksquare}}{\blacksquare}\overset{\Large{\blacksquare}}{\phantom{\blacksquare}}\\
&&&\overset{\Large{\blacksquare}}{\blacksquare}\overset{\Large{\blacksquare}}{\blacksquare}\\
&n=5&&\blacksquare\blacksquare\blacksquare\blacksquare\blacksquare\\
&&&\overset{\Large{\blacksquare}}{\blacksquare}\blacksquare\blacksquare\blacksquare&&\blacksquare\overset{\Large{\blacksquare}}{\blacksquare}\blacksquare\blacksquare\\
&&&\overset{\Large{\blacksquare}}{\blacksquare}\overset{\Large{\blacksquare}}{\blacksquare}\blacksquare&&\overset{\Large{\blacksquare}}{\blacksquare}
\blacksquare\overset{\Large{\blacksquare}}{\blacksquare}&&\blacksquare\blacksquare\overset{\Large{\blacksquare}}{\blacksquare}\overset{\Large{\blacksquare}}{\phantom{\blacksquare}}\\
&&&\overset{\Large\overset{\huge{\blacksquare}}{\blacksquare}}{\blacksquare}\blacksquare\blacksquare&&\blacksquare\overset{\Large\overset{\huge{\blacksquare}}{\blacksquare}}{\blacksquare}\blacksquare&&\overset{\Large{\blacksquare}}{\phantom{\blacksquare}}\overset{\Large\overset{\huge{\blacksquare}}{\blacksquare}}{\blacksquare}\overset{\Large{\blacksquare}}{\phantom{\blacksquare}}&&\overset{\Large\overset{\huge{\blacksquare}}{\blacksquare}}{\phantom{\blacksquare}}\overset{\Large\overset{\huge{}}{\blacksquare}}{\blacksquare}\overset{\Large\overset{}{\blacksquare}}{\phantom{\blacksquare}}&&\overset{\Large\overset{\huge{\blacksquare}}{\blacksquare}}{\phantom{\blacksquare}}\overset{\Large\overset{}{\blacksquare}}{\phantom{\blacksquare}}\overset{\Large\overset{\huge{}}{\blacksquare}}{\blacksquare}\\
&&&\overset{\Large\overset{\huge{\blacksquare}}{\blacksquare}}{\phantom{\blacksquare}}\overset{\Large\overset{\huge{}}{\blacksquare}}{\blacksquare}\blacksquare\\
\end{align*}

mathe · 发表于 2019-3-23 20:01:13

https://oeis.org/A000105
https://en.wikipedia.org/wiki/Polyomino
https://en.wikipedia.org/wiki/Pentomino

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