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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 Leonardo numbers ： ウィキペディア英語版 Leonardo number The Leonardo numbers are a sequence of numbers given by the recurrence: :$$ L(n) = \begin 1 & \mbox n = 0 \\ 1 & \mbox n = 1 \\ L(n - 1) + L(n - 2) + 1 & \mbox n > 1 \\ \end Edsger W. Dijkstra〔(EWD797 )〕 used them as an integral part of his smoothsort algorithm, and also analyzed them in some detail.〔(EWD796a )〕 Computing a second-order recurrence relation recursively and without memoization requires L(n) computations for the ''n''-th item of the series. ==Relation to Fibonacci numbers== The Leonardo numbers are related to the Fibonacci numbers by the relation $L(n)\; =\; 2\; F(n+1)\; -\; 1,\; n\; \backslash ge\; 0$. From this relation it is straightforward to derive a closed-form expression for the Leonardo numbers, analogous to Binet's formula for the Fibonacci numbers: :$L(n)\; =\; 2\; \backslash frac\}-\; 1\; =\; \backslash frac\; \backslash left(\backslash varphi^\; -\; \backslash psi^\backslash right)\; -\; 1\; =\; 2F(n+1)\; -\; 1$ where the golden ratio $\backslash varphi\; =\; \backslash left(1\; +\; \backslash sqrt\; 5\backslash right)/2$ and $\backslash psi\; =\; \backslash left(1\; -\; \backslash sqrt\; 5\backslash right)/2$ are the roots of the quadratic polynomial $x^2\; -\; x\; -\; 1\; =\; 0$. The first few Leonardo numbers are :$1,\backslash ;1,\backslash ;3,\backslash ;5,\backslash ;9,\backslash ;15,\backslash ;25,\backslash ;41,\backslash ;67,\backslash ;109,\backslash ;177,\backslash ;287,\backslash ;465,\backslash ;753,\backslash ;1219,\backslash ;1973,\backslash ;3193,\backslash ;5167,\backslash ;8361,\; \backslash ldots$ 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「Leonardo 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.