A342818 Length of the longest monochromatic arithmetic progressions of difference d in the Thue-Morse sequence (A010060).

2, 2, 8, 2, 6, 8, 8, 2, 10, 6, 4, 8, 4, 8, 20, 2, 18, 10, 5, 6, 8, 4, 6, 8, 6, 4, 11, 8, 5, 20, 32, 2, 34, 18, 4, 10, 7, 5, 9, 6, 7, 8, 8, 4, 10, 6, 6, 8, 6, 6, 9, 4, 7, 11, 9, 8, 13, 5, 9, 20, 6, 32, 68, 2, 66, 34, 6, 18, 12, 4, 9, 10, 6, 7, 9, 5, 8, 9, 11, 6

Offset

1,1

Comments

a(n) is the largest l such that there exists k with t(k) = t(k+n) = t(k+2*n) = ... = t(k+(l-1)*n), where t(n) = A010060(n).

Links

Ibai Aedo, Table of n, a(n) for n = 1..2048

Ibai Aedo, Uwe Grimm, Yasushi Nagai, and Petra Staynova, On long arithmetic progressions in binary Morse-like words, arXiv:2101.02056 [math.CO], 2021.

Ibai Aedo, Uwe Grimm, Yasushi Nagai, and Petra Staynova, Monochromatic Arithmetic Progressions in Binary Thue-Morse-Like Words, Theor. Comp. Sci. (2022).

Olga Parshina, On arithmetic index in the generalized Thue-Morse word, arXiv:1811.03884 [math.CO], 2018.

Example

For n = 3, we have t(45) = t(48) = t(51) = t(54) = t(57) = t(60) = t(63) = t(66), and no k and m>8 exist such that t(k) = t(k+3) = t(k+2*3) = ... = t(k+(m-1)*3). So a(3)=8.

Crossrefs

Cf. A010060. Starting position of the first occurrence of a monochromatic arithmetic progression of difference n and length a(n) is given by A342827.

Sequence in context: A046644 A343059 A161915 * A174354 A011147 A273168

Adjacent sequences: A342815 A342816 A342817 * A342819 A342820 A342821

Keyword

nonn

Author

Jeffrey Shallit, Mar 22 2021

Status

approved