1998/03/10 by K. Pinn, Pinn, K. · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Coding theory and cryptography #FOS: Physical sciences #Finite Group Theory Research #chao-dyn #graph theory and CDMA systems #nlin.CD
paper · pdf · doi:10.48550/arxiv.chao-dyn/9803012
Replaced to conform with version accepted by Complexity
openalex publication_date 1998/03/10 · arxiv created 1998/07/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 which have ``parents'' mostly in generation k-1, and a few from generation k-2. In this sense the series becomes Fibonacci type on a logarithmic scale. The mean square size of S(n)=Q(n)-n/2, averaged over generations is like 2**(alpha*k), with exponent alpha = 0.88(1). The probability distribution p^*(x) of x = R(n)= S(n)/n**alpha, n >> 1, is well defined and is strongly non-Gaussian. The probability distribution of xm = R(n)-R(n-m) is given by pm(xm)= lambdam * p^*(xm/lambdam). It is conjectured that lambdam goes to sqrt(2) for large m.