Order and chaos in Hofstadter'sQ(n) sequence
Abstract
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 x_m = R(n)-R(n-m) is given by p_m(x_m)= lambda_m * p^*(x_m/lambda_m). It is conjectured that lambda_m goes to sqrt(2) for large m.
- Publication:
-
Complexity
- Pub Date:
- January 1999
- DOI:
- 10.1002/(SICI)1099-0526(199901/02)4:3<41::AID-CPLX8>3.3.CO;2-V
- arXiv:
- arXiv:chao-dyn/9803012
- Bibcode:
- 1999Cmplx...4c..41P
- Keywords:
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- Nonlinear Sciences - Chaotic Dynamics
- E-Print:
- Replaced to conform with version accepted by Complexity