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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 Pell numbers ： ウィキペディア英語版 Pell number In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell-Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82. Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + √2. As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems.〔For instance, Sellers (2002) proves that the number of perfect matchings in the Cartesian product of a path graph and the graph ''K''4-''e'' can be calculated as the product of a Pell number with the corresponding Fibonacci number.〕 As with Pell's equation, the name of the Pell numbers stems from Leonhard Euler's mistaken attribution of the equation and the numbers derived from it to John Pell. The Pell-Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are Lucas sequences. == Pell numbers == The Pell numbers are defined by the recurrence relation : In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are :, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, ... . The Pell numbers can also be expressed by the closed form formula :$P\_n=\backslash frac.$ For large values of ''n'', the $\backslash scriptstyle\; (1+\backslash sqrt\; 2)^n$ term dominates this expression, so the Pell numbers are approximately proportional to powers of the silver ratio $\backslash scriptstyle\; (1+\backslash sqrt\; 2)$, analogous to the growth rate of Fibonacci numbers as powers of the golden ratio. A third definition is possible, from the matrix formula : Many identities can be derived or proven from these definitions; for instance an identity analogous to Cassini's identity for Fibonacci numbers, :$P\_P\_-P\_n^2\; =\; (-1)^n,$ is an immediate consequence of the matrix formula (found by considering the determinants of the matrices on the left and right sides of the matrix formula).〔For the matrix formula and its consequences see Ercolano (1979) and Kilic and Tasci (2005). Additional identities for the Pell numbers are listed by Horadam (1971) and Bicknell (1975).〕 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「Pell number」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.