Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site arizona.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxl!ihnp4!arizona!pete From: pete@arizona.UUCP (P. Downey) Newsgroups: net.math Subject: Re: Hoffstuff Message-ID: <14690@arizona.UUCP> Date: Tue, 21-Aug-84 23:25:33 EDT Article-I.D.: arizona.14690 Posted: Tue Aug 21 23:25:33 1984 Date-Received: Thu, 23-Aug-84 01:15:54 EDT References: <500@bunker.UUCP> Organization: Dept of CS, U of Arizona, Tucson Lines: 18 Chuck Heaton asked about the behavior of the recurrence h(0) = 0; h(k) = k - h(h(k-1)) which appears in Hofstadter's "Goedel, Escher, Bach". There is an interesting answer, details of which can be found in a paper soon to appear: Downey, P.J. and R.E. Griswold. On a family of nested recurrences. "Fibonacci Quarterly", (Nov 1984). Here's the bottom line; for complete details see the paper or mail me for a copy. Let R stand for the "golden ratio" (sqrt(5)+1)/2 and let P be its reciprocal. Let [ x ] denote the integer part of the number x, i.e., floor(x). Then h(k) = [ P*(k+1) ]. The values of the first difference function g(k) = h(k) - h(k-1) are either zero or 1. The sequence of k for which g(k)=1 form a "Beatty sequence" [R],[2*R],[3*R],...