TY - JOUR

T1 - A two-parameter family of an extension of Beatty sequences

AU - Artstein-Avidan, Shiri

AU - Fraenkel, Aviezri S.

AU - Sós, Vera T.

PY - 2008/10/28

Y1 - 2008/10/28

N2 - Beatty sequences ⌊ n α + γ ⌋ are nearly linear, also called balanced, namely, the absolute value of the difference D of the number of elements in any two subwords of the same length satisfies D ≤ 1. For an extension of Beatty sequences, depending on two parameters s, t ∈ Z> 0, we prove D ≤ ⌊ (s - 2) / (t - 1) ⌋ + 2(s, t ≥ 2), and D ≤ 2 s + 1(s ≥ 2, t = 1) . We show that each value that is assumed, is assumed infinitely often. Under the assumption (s - 2) ≤ (t - 1)2 the first result is optimal, in that the upper bound is attained. This provides information about the gap-structure of (s, t)-sequences, which, for s = 1, reduce to Beatty sequences. The (s, t)-sequences were introduced in Fraenkel [Heap games, numeration systems and sequences, Ann. Combin. 2 (1998) 197-210; E. Lodi, L. Pagli, N. Santoro (Eds.), Fun with Algorithms, Proceedings in Informatics, vol. 4, Carleton Scientific, University of Waterloo, Waterloo, Ont., 1999, pp. 99-113], where they were used to give a strategy for a 2-player combinatorial game on two heaps of tokens.

AB - Beatty sequences ⌊ n α + γ ⌋ are nearly linear, also called balanced, namely, the absolute value of the difference D of the number of elements in any two subwords of the same length satisfies D ≤ 1. For an extension of Beatty sequences, depending on two parameters s, t ∈ Z> 0, we prove D ≤ ⌊ (s - 2) / (t - 1) ⌋ + 2(s, t ≥ 2), and D ≤ 2 s + 1(s ≥ 2, t = 1) . We show that each value that is assumed, is assumed infinitely often. Under the assumption (s - 2) ≤ (t - 1)2 the first result is optimal, in that the upper bound is attained. This provides information about the gap-structure of (s, t)-sequences, which, for s = 1, reduce to Beatty sequences. The (s, t)-sequences were introduced in Fraenkel [Heap games, numeration systems and sequences, Ann. Combin. 2 (1998) 197-210; E. Lodi, L. Pagli, N. Santoro (Eds.), Fun with Algorithms, Proceedings in Informatics, vol. 4, Carleton Scientific, University of Waterloo, Waterloo, Ont., 1999, pp. 99-113], where they were used to give a strategy for a 2-player combinatorial game on two heaps of tokens.

KW - Extension of Beatty sequences

KW - Gap structure

KW - Sequences of differences

UR - http://www.scopus.com/inward/record.url?scp=48449106682&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2007.08.070

DO - 10.1016/j.disc.2007.08.070

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AN - SCOPUS:48449106682

SN - 0012-365X

VL - 308

SP - 4578

EP - 4588

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 20

ER -