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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 (list; graph; refs; listen; history; text; internal format)
 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:=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. = 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

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Last modified July 6 12:26 EDT 2022. Contains 355110 sequences. (Running on oeis4.)