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%I #11 May 26 2020 06:04:00
%S 1,1,1,2,2,2,2,2,2,3,2,2,2,2,2,3,3,3,3,3,2,2,2,3,2,2,2,3,3,3,3,3,3,4,
%T 3,3,3,3,3,4,2,2,2,3,3,3,3,3,2,2,2,3,3,3,3,3,3,4,3,3,3,3,3,4,4,4,4,4,
%U 3,3,3,4,3,3,3,4,4,4,4,4,2,2,2,3,3,3,3
%N Number of periods of the length-n prefix of the Thue-Morse sequence (A010060).
%C A period of a length-n word x is an integer p, 1 <= p <= n, such that x[i]=x[i+p] for 1 <= i <= n-p.
%C This is a 2-regular sequence. It satisfies the identity
%C a(4n) = a(n)+1 for n > 0, and the identities
%C a(4n+3) = a(4n+1)
%C a(8n+1) = a(2n+1) + t(n)
%C a(8n+2) = a(4n+1) + t(n)
%C a(8n+6) = a(4n+1) + 1-t(n)
%C a(16n+5) = a(2n+1) + 1
%C a(16n+13) = a(4n+1) + 1
%C for n >= 0. Here t(n) = A010060(n).
%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%H D. Gabric, N. Rampersad, and J. Shallit, <a href="https://arxiv.org/abs/2005.11718">An inequality for the number of periods in a word</a>, arxiv preprint arXiv:2005.11718 [cs.DM], May 24 2020.
%e For n = 10, the a(10) = 3 periods of 0110100110, the first 10 symbols of the Thue-Morse sequence, are p = 1, 4, and 10.
%Y Cf. A010060.
%K nonn
%O 1,4
%A _Jeffrey Shallit_, May 23 2020