A216958 Number of binary vectors v of length n with curling number 1 such that the concatenation v v with first term omitted also has curling number 1.

2, 2, 4, 6, 10, 20, 36, 72, 142, 280, 560, 1114, 2222, 4436, 8864, 17718, 35420, 70824, 141624, 283210, 566394, 1132728, 2265390, 4530726, 9061318, 18122518, 36244908, 72489566, 144978870, 289957490, 579914470, 1159828430, 2319656332, 4639311620, 9278622168

Offset: 1

Author

N. J. A. Sloane, Sep 27 2012

Keywords

• nonn

Comments

See A216730 for definitions.

I would very much like to have a formula or recurrence for this sequence.

Alternatively, the number of squares of length 2n over a binary alphabet having no proper prefix that is a square. Here by a square I mean a word of the form xx, where x is any word. - Jeffrey Shallit, Nov 29 2013

Examples

			Taking the alphabet to be {2,3}, v = 32232 has curling number 1, but 2232.32232 has curling number 2, so is not counted here.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..100 [Based on _Allan Wilks_'s b-file for A122536]
• Daniel Gabric, Jeffrey Shallit, Borders, Palindrome Prefixes, and Square Prefixes, arXiv:1906.03689 [cs.DM], 2019.
• Daniel Gabric, Jeffrey Shallit, Borders, palindrome prefixes, and square prefixes, Info. Proc. Letters 165 (2021), 106027.
• Index entries for sequences related to curling numbers

Crossrefs

Cf. A216730, A122536, A216959, A216960, A216961.

First column of A218875.

Formula

a(n) = 2*A122536(2n-1)-A122536(2n). - R. J. Mathar, Oct 31 2024

Extensions

a(31)-a(35) from N. J. A. Sloane, Oct 25 2012