A278815 Number of tilings of a 2 X n grid with monomers, dimers, and trimers.

1, 2, 7, 29, 109, 416, 1596, 6105, 23362, 89415, 342193, 1309593, 5011920, 19180976, 73406985, 280933906, 1075154535, 4114694797, 15747237101, 60265824784, 230641706484, 882682631025, 3378090801226, 12928199853783, 49477163668857, 189352713633433

Offset

0,2

Comments

The first three terms are the same as A030186 because there are only monomers and dimers in boards with n<3.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Kathryn Haymaker and Sara Robertson, Counting Colorful Tilings of Rectangular Arrays, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.8, Corollary 2.

Index entries for linear recurrences with constant coefficients, signature (3,2,5,-2,0,-1).

Formula

a(n) = 3*a(n-1) + 2*a(n-2) + 5*a(n-3) - 2*a(n-4) - a(n-6).

G.f.: (1 - x - x^2 - x^3)/(1 - 3*x - 2*x^2 - 5*x^3 + 2*x^4 + x^6).

Maple

seq(coeff(series((1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 28 2019

Mathematica

LinearRecurrence[{3, 2, 5, -2, 0, -1}, {1, 2, 7, 29, 109, 416}, 30] (* G. C. Greubel, Oct 28 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+ 2*x^4 +x^6)) \\ G. C. Greubel, Oct 28 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6) )); // G. C. Greubel, Oct 28 2019
(Sage)
def A278815_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6) ).list()
A278815_list(30) # G. C. Greubel, Oct 28 2019
(GAP) a:=[1, 2, 7, 29, 109, 416];; for n in [7..30] do a[n]:=3*a[n-1]+2*a[n-2] +5*a[n-3]-2*a[n-4]-a[n-6]; od; a; # G. C. Greubel, Oct 28 2019

Crossrefs

Cf. A030186, A052961, A129682, A219866.

Sequence in context: A203969 A199581 A289609 * A263367 A120757 A134169

Adjacent sequences: A278812 A278813 A278814 * A278816 A278817 A278818

Keyword

nonn,easy

Author

Kathryn Haymaker, Nov 28 2016

Status

approved