A066297 Number of ternary squarefree necklaces.

1, 3, 6, 6, 12, 0, 18, 0, 24, 0, 0, 66, 72, 78, 0, 30, 48, 0, 252, 228, 300, 42, 462, 690, 720, 750, 702, 810, 1260, 2088, 3870, 5022, 5568, 4752, 5916, 10920, 16416, 18870, 21660, 23556, 34320, 51414, 75852, 93654, 108372, 126360, 172914, 245058, 343872

Offset: 0

Author

Paul Parsons (paul.parsons6(AT)btinternet.com)

Keywords

• hard
• nice
• nonn

Comments

A square is an adjacent pair of repeats, e.g., "aa" or "abcabc". A necklace is a word that may be rotated before being tested (for squares).

Several similar sequences (with same zeros) can be constructed from equivalence classes of the loops.

Higher terms: a(n) > 0 for 30 < n <= 56; no zeros known after a(17).

This is also the number of ternary "circular" squarefree words. The Currie paper proves no 0 entries after a(17). - Jeffrey Shallit, Jul 11 2012

Examples

			a(1)=3, size of {"a","b","c"}; a(6)=18, size of {"abacbc","bacbca",...,"cbabca"}.
		

Links

• J. D. Currie, There are ternary circular square-free words of length n for n >= 18, Elect. J. Combinatorics 9 (2002), Paper N10.

Crossrefs

Variant of A006156. See also A001037, A006206.

Extensions

a(31)-a(36) from Jeffrey Shallit, Jan 22 2019

a(37)-a(48) from Sean A. Irvine, Oct 07 2023