Geodesic
Sums of digits and the distribution of generalized Thue-Morse sequences
Journal of integer sequences, Tome 20 (2017) no. 3
In this paper we study the distribution of the infinite word $t_{q,n} := (s_{q}(k)$ mod $n)_{k=0}^{\infty }$, 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., ${0, 1, \dots , n-1}$. Also, we prove some stronger related results.
Classification : 11B99, 11Y55, 11N25, 11A63, 68R15
Keywords: digit sum, thue-Morse sequence 
@article{JIS_2017__20_3_a3,
     author = {Zhao,  Hancong and Zhang,  Dong},
     title = {Sums of digits and the distribution of generalized {Thue-Morse} sequences},
     journal = {Journal of integer sequences},
     year = {2017},
     volume = {20},
     number = {3},
     zbl = {1381.11010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2017__20_3_a3/}
}
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Zhao,  Hancong; Zhang,  Dong. Sums of digits and the distribution of generalized Thue-Morse sequences. Journal of integer sequences, Tome 20 (2017) no. 3. http://geodesic.mathdoc.fr/item/JIS_2017__20_3_a3/