TAILIEUCHUNG - Báo cáo toán học: "On a Strange Recursion of Golomb"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On a Strange Recursion of Golomb. | On a Strange Recursion of Golomb Ed Barbeau Department of Mathematics University of Toronto Toronto Canada M5S 1A1 barbeau@math. utoronto. ca Steve Tanny Department of Mathematics University of Toronto Toronto Canada M5S 1A1 tanny@ Submitted January 4 1996 Accepted February 1 1996. There are patterns I must follow just as I must breathe each breath Paul Simon Patterns 1964 Abstract In an unpublished note Golomb proposed a family of strange recursions of metafibonacci type parameterized by k and for each k identified what he speculated was the unique increasing solution. We show that to the contrary there are many increasing solutions for each k and we indicate explicitly how to construct them. We also provide some additional general results concerning the nature of the strictly increasing solutions for this unusual family of recursions. Subject Number 05A11 1. Introduction In an unpublished note 1 Golomb considers a variety of sequences that satisfy strange recursions. Included among these are the well-known Hofstadter sequence 3 and the Newman-Conway sequence 5 . Golomb writes the simplest strange recursion as u u n u n and easily determines the general solution for it. Subsequently he introduces a more complicated strange recursion actually a family of recursions b b n kn 2b n kn 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 3 1996 R8 2 with initial conditions b 1 1 and b 2 3 for k 1 and b 1 1 and b 2 2 for k 1. Here k is assumed to be a positive integer. It is tempting to try a linear function in n for b n . At the same time it is necessary that b n is a positive integer so that b n kn is well-defined as an index for b. This appears to motivate the following solution given without proof by Golomb who states that one strictly increasing sequence which satisfies this recursion is nakJ where xJ is the floor of x the biggest integer less than or equal to x and ak is the positive root of the equation x2 k 2 x k 0. Golomb credits A. Fraenkel with .

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