A330131 Number of length-n binary words with no even palindrome of length > 6 and no odd palindrome of length > 3.

1, 2, 4, 8, 16, 24, 36, 52, 66, 84, 104, 128, 160, 200, 248, 308, 384, 476, 592, 740, 922, 1144, 1426, 1776, 2208, 2744, 3418, 4252, 5294, 6592, 8210, 10204, 12704, 15808, 19676, 24472, 30478, 37920, 47214, 58736, 73132, 90952, 113236, 140880, 175408, 218196

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Lukas Fleischer, Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,0,2,0,3,0,0,0,1).

Formula

a(n) = a(n - 6) + 2*a(n - 8) + 3*a(n - 10) + a(n - 14) for n >= 21. Further- more, and a(n) ~ C1*alpha^n + C2*(-alpha)^n, where C1 ~ 11.58110542, C2 ~ 0.00264754, and α ~ 1.244528319539183 is the largest real zero of X^14 - X^8 - 2X^6 - 3X^4 - 1.

G.f.: (1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 24*x^5 + 35*x^6 + 50*x^7 + 60*x^8 + 72*x^9 + 77*x^10 + 82*x^11 + 80*x^12 + 76*x^13 + 61*x^14 + 46*x^15 + 36*x^16 + 16*x^17 + 10*x^18 + 8*x^19 + 6*x^20) / (1 - x^6 - 2*x^8 - 3*x^10 - x^14). - Colin Barker, Dec 02 2019

Prog

(PARI) Vec((1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 24*x^5 + 35*x^6 + 50*x^7 + 60*x^8 + 72*x^9 + 77*x^10 + 82*x^11 + 80*x^12 + 76*x^13 + 61*x^14 + 46*x^15 + 36*x^16 + 16*x^17 + 10*x^18 + 8*x^19 + 6*x^20) / (1 - x^6 - 2*x^8 - 3*x^10 - x^14) + O(x^40)) \\ Colin Barker, Dec 02 2019

Crossrefs

Sequence in context: A333994 A305656 A005943 * A008233 A224815 A031923

Adjacent sequences: A330128 A330129 A330130 * A330132 A330133 A330134

Keyword

nonn,easy

Author

Jeffrey Shallit, Dec 02 2019

Status

approved