### Abstract

In this article, first we generalize the Thue-Morse sequence by means of a cyclic permutation and the k-adic expansion of non-negative integers, giving a sequence (a(n))_{n=0}^{∞}, and consider the condition that (a(n))_{n=0}^{∞} is non-periodic. Next, we show that, if a generalized Thue-Morse sequence (a(n))_{n=0}^{∞} is not periodic, then no subsequence of the form (a(N+nl))_{n=0}^{∞} (where N ≥ 0 and l > 0) is periodic. We apply the combinatorial transcendence criterion established by Adamczewski, Bugeaud, Luca, and Bugeaud to find that, for a non-periodic generalized Thue-Morse sequence taking its values in {0,1,...,β-1} (where β is an integer greater than 1), the series Σ_{n=0}^{∞} a(N+nl) β^{-n-1} gives a transcendental number. Furthermore, for non-periodic generalized Thue-Morse sequences taking positive integer values, the continued fraction [0, a(N), a(N+l),..., a(N+nl), ...] gives a transcendental number.

Original language | English |
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Article number | 15.9.2 |

Journal | Journal of Integer Sequences |

Volume | 18 |

Issue number | 9 |

Publication status | Published - 2015 |

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### Keywords

- Digit counting
- Stammering sequence

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Integer Sequences*,

*18*(9), [15.9.2].