Random Fibonacci sequences and the number 1.13198824

For the familiar Fibonacci sequence | deened by f 1 = f 2 = 1, and f n = f n?1 + f n?2 for n > 2 | f n increases exponentially with n at a rate given by the golden ratio (1 + p 5)=2 = 1:61803398 : : :. But for a simple modiication with both additions and subtractions | the random Fibonacci sequences deened by t 1 = t 2 = 1, and for n > 2, t n = t n?1 t n?2 , where each sign is independent and either + or ? with probability 1=2 | it is not even obvious if jt n j should increase with n. Our main result is that n p jt n j ! 1:13198824 : : : as n ! 1 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.