Random Fibonacci Sequences and the Number

For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n > 2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 = 1.61803398 . . . . But for a simple modification with both additions and subtractions — the random Fibonacci sequences defined by t1 = t2 = 1, and for n > 2, tn = ±tn−1 ± tn−2, where each ± sign is independent and either + or − with probability 1/2 — it is not even obvious if |tn| should increase with n. Our main result is that n √ |tn| → 1.13198824 . . . as n→∞ with probability 1. Finding the number 1.13198824 . . . involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.