where ,
,
and each sign is chosen independently and at random with probability 1/2. Surprisingly,
Viswanath (2000) showed that
(2)
(OEIS A078416) with probability one. This constant
is sometimes known as Viswanath's constant.
Considering the more general recurrence
(3)
the limit
(4)
exists for almost all values of . The critical value such that is given by
(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 random matrices (Bougerol and Lacrois 1985, pp. 11
and 157).
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.
News155, 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.