On generalized Stanley sequences

• Published in: Acta mathematica Hungarica Vol. 154; no. 2; pp. 501 - 510
• Main Authors: Kiss, S. Z., Sándor, Cs, Yang, Q.-H.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2018; Springer Nature B.V
• Subjects: Arithmetic; Greedy algorithms; Integers; Mathematics; Mathematics and Statistics; Stanley sequence; 11B75; arithmetic progression; probabilistic method
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-018-0791-1

Let N denote the set of all nonnegative integers. Let k ≥ 3 be an integer and A 0 = { a 1 , ⋯ , a t } ( a 1 < ⋯ < a t ) be a nonnegative set which does not contain an arithmetic progression of length k . We denote A = { a 1 , a 2 , … } defined by the following greedy algorithm: if l ≥ t and a 1 , ⋯ , a l have already been defined, then a l + 1 is the smallest integer a > a l such that { a 1 , ⋯ , a l } ∪ { a } also does not contain a k -term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A 0 . We prove some results about various generalizations of the Stanley sequence.