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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 Regular paperfolding sequence ： ウィキペディア英語版 Regular paperfolding sequence In mathematics the regular paperfolding sequence, also known as the dragon curve sequence, is an infinite automatic sequence of 0s and 1s defined as the limit of the following process: :1 :1 1 0 :1 1 0 1 1 0 0 :1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 At each stage an alternating sequence of 1s and 0s is inserted between the terms of the previous sequence. The sequence takes its name from the fact that it represents the sequence of left and right folds along a strip of paper that is folded repeatedly in half in the same direction. If each fold is then opened out to create a right-angled corner, the resulting shape approaches the dragon curve fractal. For instance the following curve is given by folding a strip four times to the right and then unfolding to give right angles, this gives the first 15 terms of the sequence when 1 represents a right turn and 0 represents a left turn. Starting at ''n'' = 1, the first few terms of the regular paperfolding sequence are: :1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, ... ==Properties== The value of any given term ''t''''n'' in the regular paperfolding sequence can be found recursively as follows. If ''n'' = ''m''·2''k'' where ''m'' is odd then :$t\_n\; =$ \begin 1 & \text m = 1 \mod 4 \\ 0 & \text m = 3 \mod 4 \end Thus ''t''12 = ''t''3 = 0 but ''t''13 = 1. The paperfolding word 1101100111001001..., which is created by concatenating the terms of the regular paperfolding sequence, is a fixed point of the morphism or string substitution rules :11 → 1101 :01 → 1001 :10 → 1100 :00 → 1000 as follows: :11 → 1101 → 11011001 → 1101100111001001 → 11011001110010011101100011001001 ... It can be seen from the morphism rules that the paperfolding word contains at most three consecutive 0s and at most three consecutive 1s. The paperfolding sequence also satisfies the symmetry relation: :$t\_n\; =$ \begin 1 & \text n = 2^k \\ 1-t_ & \text 2^\end which shows that the paperfolding word can be constructed as the limit of another iterated process as follows: :1 :1 1 0 :110 1 100 :1101100 1 1100100 :110110011100100 1 110110001100100 In each iteration of this process, a 1 is placed at the end of the previous iteration's string, then this string is repeated in reverse order, replacing 0 by 1 and vice versa. 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「Regular paperfolding 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.