A342512 a(n) is the number of substrings of the binary representation of n that are instances of the Zimin word Z_k, where k = A342510(n).

1, 1, 3, 3, 6, 1, 6, 1, 1, 1, 2, 2, 10, 2, 1, 3, 3, 2, 4, 2, 3, 4, 4, 4, 1, 2, 3, 4, 1, 4, 3, 6, 6, 4, 6, 3, 6, 6, 5, 4, 5, 6, 7, 6, 5, 8, 6, 7, 3, 3, 5, 4, 4, 6, 6, 7, 2, 4, 5, 7, 3, 7, 6, 10, 10, 7, 9, 5, 10, 8, 7, 5, 9, 9, 10, 8, 8, 9, 7, 7, 8, 8, 11, 8, 9

Offset

0,3

Comments

This value of k is chosen so that Z_k is the largest Zimin word that the binary expansion of n does not avoid.

Links

Peter Kagey, Table of n, a(n) for n = 0..8191

Peter Kagey, Matching ABACABA-type patterns, Code Golf Stack Exchange.

Danny Rorabaugh, Toward the Combinatorial Limit Theory of Free Words, arXiv preprint arXiv:1509.04372 [math.CO], 2015.

Wikipedia, Sesquipower.

Formula

a(n) = A342511(n, A342510(n)).

Example

For n = 121, the binary expansion is "1111001", which avoids the Zimin word Z_3 = ABACABA, but does not avoid the Zimin word Z_2 = ABA. In particular, there are a(121) = 7 substrings that are instances of Z_2:
(111)1001 with A = 1 and B = 1,
1(111)001 with A = 1 and B = 1,
(1111)001 with A = 1 and B = 11,
111(1001) with A = 1 and B = 00,
11(11001) with A = 1 and B = 100,
1(111001) with A = 1 and B = 1100, and
(1111001) with A = 1 and B = 11100.

Crossrefs

Cf. A342510, A342511.

Sequence in context: A327824 A189915 A085572 * A205548 A010609 A066519

Adjacent sequences: A342509 A342510 A342511 * A342513 A342514 A342515

Keyword

nonn,base,look

Author

Peter Kagey, Mar 14 2021

Status

approved