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In mathematics, the look-and-say sequence is the sequence of integers beginning as follows: : 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... . To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example: * 1 is read off as "one 1" or 11. * 11 is read off as "two 1s" or 21. * 21 is read off as "one 2, then one 1" or 1211. * 1211 is read off as "one 1, then one 2, then two 1s" or 111221. * 111221 is read off as "three 1s, then two 2s, then one 1" or 312211. The look-and-say sequence was introduced and analyzed by John Conway. The idea of the look-and-say sequence is similar to that of run-length encoding. If we start with any digit ''d'' from 0 to 9 then ''d'' will remain indefinitely as the last digit of the sequence. For ''d'' different from 1, the sequence starts as follows: : ''d'', 1''d'', 111''d'', 311''d'', 13211''d'', 111312211''d'', 31131122211''d'', … Ilan Vardi has called this sequence, starting with ''d'' = 3, the Conway sequence . (for ''d'' = 2, see )〔(Conway Sequence ), MathWorld, accessed on line February 4, 2011.〕 == Basic properties == * The sequence grows indefinitely. In fact, any variant defined by starting with a different integer seed number will (eventually) also grow indefinitely, except for the degenerate sequence: 22, 22, 22, 22, … 〔 * No digits other than 1, 2, and 3 appear in the sequence, unless the seed number contains such a digit or a run of more than three of the same digit.〔 〕 * Conway's cosmological theorem: Every sequence eventually splits into a sequence of "atomic elements", which are finite subsequences that never again interact with their neighbors. There are 92 elements containing the digits 1, 2, and 3 only, which John Conway named after the natural chemical elements. There are also two "transuranic" elements for each digit other than 1, 2, and 3.〔〔Ekhad, S. B., Zeilberger, D.: (Proof of Conway's lost cosmological theorem ), Electronic Research Announcements of the American Mathematical Society, August 21, 1997, Vol. 5, pp. 78-82. Retrieved July 4, 2011.〕 * The terms eventually grow in length by about 30% per generation. In particular, if ''L''''n'' denotes the number of digits of the ''n''-th member of the sequence, then the limit of the ratio exists and is given by :: : where λ = 1.303577269034... is an algebraic number of degree 71.〔 This fact was proven by Conway, and the constant λ is known as Conway's constant. The same result also holds for every variant of the sequence starting with any seed other than 22. Conway's constant is the unique positive real root of the following polynomial: : 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「Look-and-say sequence」の詳細全文を読む スポンサード リンク
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