Order and chaos in Hofstadter's Q(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2^k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2^αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/n^α, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of x_m = R(n) − R(n − m) is given by p_m(x_m) = λ_m p*(x_m/λ_m), with λ_m → √2 for large m. © 1999 John Wiley & Sons, Inc.