A two-parameter family of an extension of Beatty sequences

Abstract

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.