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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],...