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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 Ducci sequence ： ウィキペディア英語版 Ducci sequence A Ducci sequence is a sequence of ''n''-tuples of integers, sometimes known as "the Diffy game", because it is based on sequences. Given an ''n''-tuple of integers $(a\_1,a\_2,...,a\_n)$, the next ''n''-tuple in the sequence is formed by taking the absolute differences of neighbouring integers: :$(a\_1,a\_2,...,a\_n)\; \backslash rightarrow\; (|a\_1-a\_2|,\; |a\_2-a\_3|,\; ...,\; |a\_n-a\_1|)\backslash ,\; .$ Another way of describing this is as follows. Arrange ''n'' integers in a circle and make a new circle by taking the difference between neighbours, ignoring any minus signs; then repeat the operation. Ducci sequences are named after Enrico Ducci, the Italian mathematician credited with their discovery. Ducci sequences are also known as the Ducci map or the n-number game. Open problems in the study of these maps still remain. ==Properties== From the second ''n''-tuple onwards, it is clear that every integer in each ''n''-tuple in a Ducci sequence is greater than or equal to 0 and is less than or equal to the difference between the maximum and mimimum members of the first ''n''-tuple. As there are only a finite number of possible ''n''-tuples with these constraints, the sequence of n-tuples must sooner or later repeat itself. Every Ducci sequence therefore eventually becomes periodic. If ''n'' is a power of 2 every Ducci sequence eventually reaches the ''n''-tuple (0,0,...,0) in a finite number of steps.〔 If ''n'' is ''not'' a power of two, a Ducci sequence will either eventually reach an ''n''-tuple of zeros or will settle into a periodic loop of 'binary' ''n''-tuples; that is, ''n''-tuples which contain only two different digits. An obvious generalisation of Ducci sequences is to allow the members of the ''n''-tuples to be any real numbers rather than just integers. The properties presented here do not always hold for these generalisations. For example, a Ducci sequence starting with the ''n''-tuple (1, ''q'', ''q''2, ''q''3) where ''q'' is the (irrational) positive root of the cubic $x^3\; -\; x^2\; -\; x\; -\; 1\; =\; 0$ does not reach (0,0,0,0) in a finite number of steps, although in the limit it converges to (0,0,0,0).〔 〕 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「Ducci 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.