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Beatty Sequences, Continued Fractions, and Certain Shift Operators

Published online by Cambridge University Press:  20 November 2018

Kenneth B. Stolarsky
Kenneth B. Stolarsky*
Affiliation:
Department of Mathematics, University of ColoradoBoulder, Colorado80302
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Abstract

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Let θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.

Keywords

10A3510A3010F20Beatty sequencescomplementary sequences of natural numberscontinued fractionsFibonacci sequencegolden meangreatest integer functionshift operatorsWythoff’s game
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 4 , 01 December 1976 , pp. 473 - 482
Copyright
Copyright © Canadian Mathematical Society 1976

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