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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 Somos sequence ： ウィキペディア英語版 Somos sequence In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. It is not obvious from the form of their defining recurrence that every number in a Somos sequence is an integer, but nevertheless many Somos sequences have the property that all of their members are integers. ==Recurrence equations== For an integer number ''k'' larger than 1, the Somos-''k'' sequence $(a\_0,\; a\_1,\; a\_2,\; \backslash ldots\; )$ is defined by the equation :$a\_n\; a\_\; =\; a\_\; a\_\; +\; a\_\; a\_\; +\; \backslash cdots\; +\; a\_\; a\_$ when ''k'' is odd, or by the analogous equation :$a\_n\; a\_\; =\; a\_\; a\_\; +\; a\_\; a\_\; +\; \backslash cdots\; +\; (a\_)^2$ when ''k'' is even, together with the initial values : ''a''''i'' = 1 for ''i'' < ''k''. For ''k'' = 2 or 3, these recursions are very simple (there is no addition on the right-hand side) and they define the all-ones sequence (1, 1, 1, 1, 1, 1, ...). In the first nontrivial case, ''k'' = 4, the defining equation is :$a\_n\; a\_\; =\; a\_\; a\_\; +\; a\_^2$ while for ''k'' = 5 the equation is :$a\_n\; a\_\; =\; a\_\; a\_\; +\; a\_\; a\_\backslash ,.$ These equations can be rearranged into the form of a recurrence relation, in which the value ''a''''n'' on the left hand side of the recurrence is defined by a formula on the right hand side, by dividing the formula by ''a''''n'' − ''k''. For ''k'' = 4, this yields the recurrence :$a\_n\; =\; \backslash frac\; +\; a\_^2\}\; a\_\; +\; a\_\; a\_\}\{a\_\{n-5\}\}.$ While in the usual definition of the Somos sequences, the values of ''a''''i'' for ''i'' < ''k'' are all set equal to 1, it is also possible to define other sequences by using the same recurrences with different initial values. 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「Somos sequence」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.