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Sums of Digits and the Distribution of Generalized Thue-Morse Sequences
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Hancong Zhao and Dong Zhang

Peking University

Beijing 100871

P. R. China

**Abstract:**

In this paper we study the distribution of the infinite word
*t*_{q,n} :=
(*s*_{q}(*k*) mod
*n*)_{k=0}^{∞},
which we call the generalized Thue-Morse sequence.
Here *s*_{q}(*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 *t*_{q,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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