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ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

Published online by Cambridge University Press:  01 May 2009

ARTŪRAS DUBICKAS
ARTŪRAS DUBICKAS*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania E-mail: arturas.dubickas@mif.vu.lt
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Abstract

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Let x0 < x1 < x2 < ⋅⋅⋅ be an increasing sequence of positive integers given by the formula xn=⌊βxn−1 + γ⌋ for n=1, 2, 3, . . ., where β > 1 and γ are real numbers and x0 is a positive integer. We describe the conditions on integers bd, . . ., b0, not all zero, and on a real number β > 1 under which the sequence of integers wn=bdxn+d + ⋅⋅⋅ + b0xn, n=0, 1, 2, . . ., is bounded by a constant independent of n. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequence qxn+1 − pxn ∈ {0, 1, . . ., q−1}, n=0, 1, 2, . . ., where x0 is a positive integer, p > q > 1 are coprime integers and xn=⌈pxn−1/q⌉ for n=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 or S:x↦(3x+1)/2) in the 3x+1 problem is also given.

Keywords

11B5011B8311R0668R15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 2 , May 2009 , pp. 243 - 252
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

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