A330130 Number of length-n binary words having no even palindromes of length > 2 and no odd palindromes of length > 5.

1, 2, 4, 8, 12, 18, 26, 28, 30, 32, 34, 36, 40, 44, 48, 52, 56, 60, 64, 68, 74, 80, 88, 96, 104, 112, 120, 128, 138, 148, 162, 176, 192, 208, 224, 240, 258, 276, 300, 324, 354, 384, 416, 448, 482, 516, 558, 600, 654, 708, 770, 832, 898, 964, 1040, 1116, 1212

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 (0,0,0,0,0,0,0,1,0,1).

Formula

a(n) = a(n - 8)+a(n - 10) for n >= 16.

Furthermore, a(n) ~ C1*alpha^n +C2*(-alpha)^n, C1 ~ 15.991809, C2 ~ 0.023895, and α ~ 1.0804184273981 is the largest real zero of X^10 - X^2 -1.

G.f.: (1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 18*x^5 + 26*x^6 + 28*x^7 + 29*x^8 + 30*x^9 + 29*x^10 + 26*x^11 + 24*x^12 + 18*x^13 + 10*x^14 + 6*x^15) / (1 - x^8 - x^10). - Colin Barker, Dec 02 2019

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1, 0, 1}, {1, 2, 4, 8, 12, 18, 26, 28, 30, 32, 34, 36, 40, 44, 48, 52}, 60] (* Harvey P. Dale, Oct 15 2021 *)

Prog

(PARI) Vec((1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 18*x^5 + 26*x^6 + 28*x^7 + 29*x^8 + 30*x^9 + 29*x^10 + 26*x^11 + 24*x^12 + 18*x^13 + 10*x^14 + 6*x^15) / (1 - x^8 - x^10) + O(x^60)) \\ Colin Barker, Dec 02 2019

Crossrefs

Sequence in context: A256885 A293495 A375978 * A053799 A343949 A284122

Adjacent sequences: A330127 A330128 A330129 * A330131 A330132 A330133

Keyword

nonn,easy

Author

Jeffrey Shallit, Dec 02 2019

Status

approved