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

Author

Jeffrey Shallit, Mar 22 2021

Keywords

• nonn

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).

Examples

			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.
		

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.

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.