A232621 The number of vertically fault-free domino tilings of the 5 X (2n) board.

1, 8, 31, 175, 1015, 5911, 34447, 200767, 1170151, 6820135, 39750655, 231683791, 1350352087, 7870428727, 45872220271, 267362892895, 1558305137095, 9082467929671, 52936502440927, 308536546715887, 1798282777854391, 10481160120410455, 61088677944608335

Offset: 0

Author

R. J. Mathar, Nov 27 2013

Keywords

• nonn
• easy

Comments

A003775 counts the tilings of the 5 X (2n) board, and this sequence here counts only those that cannot be broken into tilings of two or more smaller 5 X (2n') boards with edge lengths n' < n by cutting "vertically" through the tiling parallel to the "short" side of length 5.

Technically speaking this is the inverse INVERT transform of A003775 (see the comment in A005178).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Cf. A003775, A068924.

Programs

PARI

Vec((-18*x^2+13*x^3-x^4+x+1)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 05 2016

Formula

G.f.: (1 + x - 18*x^2 + 13*x^3 - x^4)/((1-x)*(1 - 6*x + x^2)).

a(n) = 1 + 6*A001653(n) for n>1. - Bruno Berselli, Nov 27 2013

a(n) = 6*a(n-1) - a(n-2) - 4, n>=4. - R. J. Mathar, Nov 07 2015

a(n) = 1 + (3/2)*(3-2*sqrt(2))^n*(2+sqrt(2)) + (3-3/sqrt(2))*(3+2*sqrt(2))^n for n>1. - Colin Barker, Mar 05 2016