A330878 Number of solutions of length n to the word equation X_1^2 ... X_n^2 = (X_1 ... X_n)^2 in the language of optimal squareful words.

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 13, 14, 14, 15, 15, 16, 16, 17, 19, 18, 18, 20, 19, 20, 21, 22, 21, 24, 22, 23, 24, 24, 24, 27, 25, 26, 30, 27, 27, 30, 30, 32, 33, 30, 30, 35, 31, 32, 33, 33, 34, 38, 34, 35, 43

Offset

1,2

Comments

The solutions are counted up to the isomorphism 0 <-> 1 and the operation that exchanges the first two letters of a word.

Links

Table of n, a(n) for n=1..69.

J. Peltomäki and A. Saarela, Standard words and solutions of the word equation X_1^2 .. X_n^2 = (X_1 .. X_n)^2, arXiv preprint arXiv:2004.14657 [cs.FL], 2020.

J. Peltomäki and M. A. Whiteland, A square root map on Sturmian words, The Electronic Journal of Combinatorics, Vol. 24.1 #P1.54 (2017).

Formula

a(n) = floor(n/2) + 1 + Sum_{d|n, d > 2} (2^(A000374(n/d) - 1) - 1)*(A000010(d)/2 - A000005(d-1) + 1).

Example

01010010 is a solution with X_1 = 01, X_2 = 0, X_3 = 10010. Other solutions of length 8 (up to isomorphism and exchange of first two letters) are 00000000, 01000000, 01000100, 01010101.

Prog

(PARI) f(n) = {sumdiv(n >> valuation(n, 2), d, eulerphi(d)/znorder(Mod(2, d)))}; \\ A000374
a(n) = n\2 + 1 + sumdiv(n, d, if (d>2, (2^(f(n/d) - 1) - 1)*(eulerphi(d)/2 - numdiv(d-1) + 1))); \\ Michel Marcus, Apr 30 2020

Crossrefs

Cf. A000005, A000010, A000374.

Sequence in context: A110654 A350899 A350898 * A350894 A109728 A327036

Adjacent sequences: A330875 A330876 A330877 * A330879 A330880 A330881

Keyword

nonn

Author

Jarkko Peltomäki, Apr 30 2020

Status

approved