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In mathematics the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence is an infinite automatic sequence named after Marcel Golay, Walter Rudin and Harold S. Shapiro, who independently investigated its properties. ==Definition== Each term of the Rudin–Shapiro sequence is either +1 or −1. The ''n''th term of the sequence, ''b''''n'', is defined by the rules: : : where the ε''i'' are the digits in the binary expansion of ''n''. Thus ''a''''n'' counts the number of (possibly overlapping) occurrences of the sub-string 11 in the binary expansion of ''n'', and ''b''''n'' is +1 if ''a''''n'' is even and −1 if ''a''''n'' is odd.〔〔Everest et al (2003) p.234〕 For example, ''a''6 = 1 and ''b''6 = −1 because the binary representation of 6 is 110, which contains one occurrence of 11; whereas ''a''7 = 2 and ''b''7 = +1 because the binary representation of 7 is 111, which contains two (overlapping) occurrences of 11. Starting at ''n'' = 0, the first few terms of the ''a''''n'' sequence are: :0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ... and the corresponding terms ''b''''n'' of the Rudin–Shapiro sequence are: :+1, +1, +1, −1, +1, +1, −1, +1, +1, +1, +1, −1, −1, −1, +1, −1, ... 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「Rudin–Shapiro sequence」の詳細全文を読む スポンサード リンク
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