For any fixed
and
(not both equal to zero), define as the mean value of the th term of a random
Fibonacci sequence starting from and . Then the ratio tends to a constant , where has the value
(1)
(2)
(OEIS A137421; Rittaud 2007, Janvresse et al. 2008, Finch 2024), where denotes the first (and in this case only real) root
of the polynomial.
Finch, S. R. "Errata and Addenda to Mathematical Constants." 27 May 2024. https://arxiv.org/abs/2001.00578.Janvresse,
E.; Rittaud, B.; and de la Rue, T. "Growth Rate for the Expected Value of a
Generalized Random Fibonacci Sequence." 15 Apr 2008. https://arxiv.org/abs/0804.2400.Rittaud,
B. "On the Average Growth of Random Fibonacci Sequences." J. Integer
Seq.10, Article 07.2.4, 2007. https://cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/rittaud11.html.Rittaud,
B.; Janvresse, E.; Lesigne, E. and Novelli, J.-C. Quand les maths se font discrètes.
Le Pommier, p. 119, 2008.Sloane, N. J. A. Sequence 137421
A
and 
(not both equal to zero), define 
as the mean value of the 
th term of a random
Fibonacci sequence starting from 
and 
. Then the ratio 
tends to a constant 
, where 
has the value
denotes the first (and in this case only real) root
of the polynomial 
.
