On the growth of Stanley sequences

نویسندگان

  • David Rolnick
  • Praveen S. Venkataramana
چکیده

A set is said to be 3-free if no three elements form an arithmetic progression. Given a 3-free set A of integers 0 = a0 < a1 < · · · < at, the Stanley sequence S(A) = {an} is defined using the greedy algorithm: For each successive n > t, we pick the smallest possible an so that {a0, a1, . . . , an} is 3-free and increasing. Work by Odlyzko and Stanley indicates that Stanley sequences may be divided into two classes. Sequences of Type 1 are highly structured and satisfy αn2 /2 ≤ an ≤ αn log 2 , for some constant α, while those of Type 2 are chaotic and satisfy Θ(n/ logn). In this paper, we consider the possible values for α in the growth of Type 1 Stanley sequences. Whereas Odlyzko and Stanley assumed α = 1, we show that α can be any rational number which is at least 1 and for which the denominator, in lowest terms, is a power of 3.

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عنوان ژورنال:
  • Discrete Mathematics

دوره 338  شماره 

صفحات  -

تاریخ انتشار 2015