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A278815
Number of tilings of a 2 X n grid with monomers, dimers, and trimers.
1
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
Kathryn Haymaker and Sara Robertson, Counting Colorful Tilings of Rectangular Arrays, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.8, Corollary 2.
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
KEYWORD
nonn,easy
AUTHOR
Kathryn Haymaker, Nov 28 2016
STATUS
approved