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A335108 Number of periods of the length-n prefix of the Thue-Morse sequence (A010060). 0
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, 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, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 4, 2, 2, 2, 3, 3, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

This is a 2-regular sequence. It satisfies the identity

a(4n) = a(n)+1 for n > 0, and the identities

a(4n+3) = a(4n+1)

a(8n+1) = a(2n+1) + t(n)

a(8n+2) = a(4n+1) + t(n)

a(8n+6) = a(4n+1) + 1-t(n)

a(16n+5) = a(2n+1) + 1

a(16n+13) = a(4n+1) + 1

for n >= 0.   Here t(n) = A010060(n).

LINKS

Table of n, a(n) for n=1..87.

Index entries for 2-automatic sequences.

D. Gabric, N. Rampersad, and J. Shallit, An inequality for the number of periods in a word, arxiv preprint arXiv:2005.11718 [cs.DM], May 24 2020.

EXAMPLE

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.

CROSSREFS

Cf. A010060.

Sequence in context: A024708 A096917 A336664 * A320011 A068049 A297850

Adjacent sequences:  A335105 A335106 A335107 * A335109 A335110 A335111

KEYWORD

nonn

AUTHOR

Jeffrey Shallit, May 23 2020

STATUS

approved

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Last modified August 2 08:20 EDT 2021. Contains 346422 sequences. (Running on oeis4.)