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隨機斐波那契數列

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ViswanathsConstant

考慮類斐波那契遞推關係

a_n=+/-a_(n-1)+/-a_(n-2),
(1)

其中 a_0=0, a_1=1, 並且每個符號以 1/2 的機率獨立且隨機地選擇。令人驚訝的是，Viswanath (2000) 證明了

lim_(n->infty)|a_n|^(1/n)=1.13198824...
(2)

(OEIS A078416) 機率為 1。這個常數有時被稱為 Viswanath 常數。

考慮更一般的遞推關係

x_(n+1)=x_n+/-betax_(n-1),
(3)

極限

sigma(beta)=lim_(n->infty)|x_n|^(1/n)
(4)

對於幾乎所有 beta 的值都存在。臨界值 beta^* 使得 sigma(beta^*)=1 由下式給出

beta^*=0.70258...
(5)

(OEIS A118288) 並且有時被稱為 Embree-Trefethen 常數。

由於斐波那契數可以計算為 Fibonacci Q-矩陣 的乘積，因此這個相同的常數出現在某些 2×2 隨機矩陣 的迭代乘法中 (Bougerol and Lacrois 1985, pp. 11 和 157)。

另請參閱

斐波那契數, 隨機矩陣

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參考文獻

Batista Oliveira, J. and De Figueiredo, L. H. "Interval Computation of Viswanath's Constant." Reliab. Comput. 8, 131-138, 2002.Bougerol, P. and Lacrois, J. Random Products of Matrices With Applications to Infinite-Dimensional Schrödinger Operators. Basel, Switzerland: Birkhäuser, 1985.Devlin, K. "Devlin's Angle: New Mathematical Constant Discovered: Descendent of Two Thirteenth Century Rabbits." March 1999. http://www.maa.org/devlin/devlin_3_99.html.Embree, M. and Trefethen, L. N. "Growth and Decay of Random Fibonacci Sequences." Roy. Soc. London Proc. Ser. A, Math. Phys. Eng. Sci. 455, 2471-2485, 1999.Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 227-228, 2002.Michon, G. P. "Final Answers: Numerical Constants." http://home.att.net/~numericana/answer/constants.htm#viswanath.Peterson, I. "Fibonacci at Random: Uncovering a New Mathematical Constant." Sci. News 155, 376, June 12, 1999. http://sciencenews.org/sn_arc99/6_12_99/bob1.htm.Sloane, N. J. A. Sequences A078416 and A118288 in "The On-Line Encyclopedia of Integer Sequences."Viswanath, D. "Random Fibonacci Sequences and the Number 1.13198824...." Math. Comput. 69, 1131-1155, 2000.

在 中被引用

隨機斐波那契數列

請引用為

Weisstein, Eric W. "Random Fibonacci Sequence." 來自 -- Wolfram 網路資源. https://mathworld.tw/RandomFibonacciSequence.html

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