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https:///mymemo.xyz/wiki/api.php?action=feedcontributions&feedformat=atom&user=110.66.150.163 2024-06-22T04:00:17Z 利用者の投稿記録 MediaWiki 1.31.0 2015-06-29T07:41:48Z

<p>110.66.150.163: /* 関連項目 */</p> <hr /> <div>&#039;&#039;&#039;ペル数&#039;&#039;&#039;（ぺるすう、&#039;&#039;Pell number&#039;&#039;）は[[自然数]]で、n番目のペル数を P&lt;sub&gt;n&lt;/sub&gt; とおいて以下の式で定義される[[数列]]にある項のことである。<br /> :&lt;math&gt;P_1 = 1 , P_2 = 2 \,&lt;/math&gt;<br /> :&lt;math&gt;P_n = 2P_{n-1} + P_{n-2} \quad (n \ge 3)&lt;/math&gt;<br /> ペル数を1から小さい順に列記すると<br /> :[[1]], [[2]], [[5]], [[12]], [[29]], [[70]], [[169]], [[408]], [[985]], 2378, 5741, …<br /> ペル数は前項を2倍した数と前々項との和になっている。なお0番目のペル数を0と定義する場合もある。<br /> <br /> n番目のペル数は<br /> :&lt;math&gt;P_n=\frac{(1+\sqrt2)^n-(1-\sqrt2)^n}{2\sqrt2}&lt;/math&gt;<br /> という式で表される。&lt;math&gt;\scriptstyle \left\vert 1-\sqrt 2 \right\vert &lt;1&lt;/math&gt; であるため、nが大きくなるにつれて隣接するペル数の比 P&lt;sub&gt;n+1&lt;/sub&gt;/P&lt;sub&gt;n&lt;/sub&gt; は[[白銀比|白銀数]] &lt;math&gt;\scriptstyle 1+\sqrt 2&lt;/math&gt; に限りなく近付く。<br /> <br /> [[行列]]では以下のように表現される。<br /> :&lt;math&gt;\begin{pmatrix} P_{n+1} &amp; P_n \\ P_n &amp; P_{n-1} \end{pmatrix} = \begin{pmatrix} 2 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}^n&lt;/math&gt;<br /> <br /> ここから以下の[[恒等式]]が導かれる。<br /> :&lt;math&gt;P_{n+1}P_{n-1}-P_n^2 = (-1)^n&lt;/math&gt;<br /> この式はペル数を[[フィボナッチ数]]に入れ替えても当てはまる。<br /> <br /> &lt;math&gt;\displaystyle x^2-2y^2=\pm 1&lt;/math&gt;　の自然数解 x,y を小さい順に並べるとyはペル数となる。またその x/y の値は<br /> :&lt;math&gt;\frac{P_{n-1}+P_n}{P_n} = \frac11, \frac32, \frac75, \frac{17}{12}, \frac{41}{29}, \frac{99}{70}, \dots&lt;/math&gt;　とだんだん[[√2]]の値に近付く。<br /> <br /> ペル数の内[[累乗数]]は1と169のみである。<br /> <br /> ペル数を使った以下の式で[[平方三角数]]を計算できる。<br /> :&lt;math&gt;\bigl((P_{k-1}+P_k)\cdot P_k\bigr)^2 = \frac{(P_{k-1}+P_k)^2\cdot\left((P_{k-1}+P_k)^2-(-1)^k\right)}{2}.&lt;/math&gt;<br /> 左辺は[[平方数]]、右辺は[[三角数]]を表している。<br /> <br /> また以下の式で a&lt;sup&gt;2&lt;/sup&gt;+b&lt;sup&gt;2&lt;/sup&gt;=c&lt;sup&gt;2&lt;/sup&gt; を満たす[[ピタゴラス数]]を表すこともできる。<br /> :&lt;math&gt;(a,b,c)=(2P_{n}P_{n+1} , P_{n+1}^2 - P_{n}^2 , P_{n+1}^2 + P_{n}^2=P_{2n+1})&lt;/math&gt;<br /> <br /> == 関連項目 ==<br /> * [[フィボナッチ数]]<br /> * [[白銀比]]<br /> * [[平方三角数]]<br /> * [[ピタゴラス数]]<br /> * [[ペル方程式]] - [[ジョン・ペル]]<br /> <br /> {{級数}}<br /> <br /> {{DEFAULTSORT:へるすう}}<br /> [[Category:数論]]<br /> [[Category:整数の類]]<br /> [[Category:数学に関する記事]]</div>

110.66.150.163
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