Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.3

Sums of Digits and the Distribution of Generalized Thue-Morse Sequences

Hancong Zhao and Dong Zhang
Peking University
Beijing 100871
P. R. China


In this paper we study the distribution of the infinite word tq,n := (sq(k) mod n)k=0, which we call the generalized Thue-Morse sequence. Here sq(k) is the digit sum of k in base q. We give an explicit formulation of the exact minimal value of M such that every M consecutive terms in tq,n cover the residue system of n, i.e., {0, 1, ... , n-1}. Also, we prove some stronger related results.

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(Concerned with sequences A010060 A141803.)

Received April 12 2016; revised version received July 24 2016; August 31 2016; January 7 2017. Published in Journal of Integer Sequences, January 7 2017.

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