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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Square triangular number ： ウィキペディア英語版 Square triangular number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are an infinite number of square triangular numbers; the first few are 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 . ==Explicit formulas== Write ''N''''k'' for the ''k''th square triangular number, and write ''s''''k'' and ''t''''k'' for the sides of the corresponding square and triangle, so that :$N\_k\; =\; s\_k^2\; =\; \backslash frac.$ Define the ''triangular root'' of a triangular number $N\; =\; \backslash frac$ to be $n$. From this definition and the quadratic formula, $n\; =\; \backslash frac.$ Therefore, $N$ is triangular if and only if $8N\; +\; 1$ is square, and naturally $N^2$ is square and triangular if and only if $8N^2\; +\; 1$ is square, i. e., there are numbers $x$ and $y$ such that $x^2\; -\; 8y^2\; =\; 1$. This is an instance of the Pell equation, with n=8. All Pell equations have the trivial solution (1,0), for any n; this solution is called the zeroth, and indexed as $(x\_0,y\_0)$. If $(x\_k,y\_k)$ denotes the k'th non-trivial solution to any Pell equation for a particular n, it can be shown by the method of descent that $x\_\; =\; 2x\_k\; x\_1\; -\; x\_$ and $y\_\; =\; 2y\_k\; x\_1\; -\; y\_$. Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever n is not a square. The first non-trivial solution when n=8 is easy to find: it is (3,1). A solution $(x\_k,y\_k)$ to the Pell equation for n=8 yields a square triangular number and its square and triangular roots as follows: $s\_k\; =\; y\_k\; ,\; t\_k\; =\; \backslash frac,$ and $N\_k\; =\; y\_k^2.$ Hence, the first square triangular number, derived from (3,1), is 1, and the next, derived from (17,6) (=6×(3,1)-(1,0)), is 36. The sequences ''N''''k'', ''s''''k'' and ''t''''k'' are the OEIS sequences , , and respectively. In 1778 Leonhard Euler determined the explicit formula〔 〕〔 〕 :$N\_k\; =\; \backslash left(\; \backslash frac)^k\}$ N_k &= \left( ( 1 + \sqrt )^ - ( 1 - \sqrt )^ \right)^2 = \left( ( 1 + \sqrt )^-2 + ( 1 - \sqrt )^ \right) \\ &= \left( ( 17 + 12\sqrt )^k -2 + ( 17 - 12\sqrt )^k \right). \end The corresponding explicit formulas for ''s''''k'' and ''t''''k'' are 〔 :$s\_k\; =\; \backslash frac)^k\})^k\; +\; (3\; -\; 2\backslash sqrt)^k\; -\; 2\}.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Square triangular number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.