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Random Fibonacci Sequence

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ViswanathsConstant

Consider the Fibonacci-like recurrence

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

where a_0=0, a_1=1, and each sign is chosen independently and at random with probability 1/2. Surprisingly, Viswanath (2000) showed that

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

(OEIS A078416) with probability one. This constant is sometimes known as Viswanath's constant.

Considering the more general recurrence

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

the limit

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

exists for almost all values of beta. The critical value beta^* such that sigma(beta^*)=1 is given by

beta^*=0.70258...
(5)

(OEIS A118288) and is sometimes known as the Embree-Trefethen constant.

Since Fibonacci numbers can be computed as products of Fibonacci Q-matrices, this same constant arises in the iterated multiplication of certain pairs of 2×2 random matrices (Bougerol and Lacrois 1985, pp. 11 and 157).

See also

Fibonacci Number, Random Matrix

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References

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.

Referenced on Wolfram|Alpha

Random Fibonacci Sequence

Cite this as:

Weisstein, Eric W. "Random Fibonacci Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RandomFibonacciSequence.html

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