A141344
Expansion of (2-sqrt(1+4x))/(2-x-sqrt(1+4x)).
Original entry on oeis.org
1, 1, 3, 7, 19, 45, 123, 285, 807, 1771, 5407, 10587, 37627, 57619, 279783, 231615, 2307339, -387531, 21769251, -28249347, 235837791, -539858235, 2857845723, -8509970007, 37342507167, -126289733319, 510715973643, -1837291760147
Offset: 0
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CoefficientList[Series[(2-Sqrt[1+4x])/(2-x-Sqrt[1+4x]),{x,0,30}],x] (* Harvey P. Dale, Jan 14 2013 *)
A278815
Number of tilings of a 2 X n grid with monomers, dimers, and trimers.
Original entry on oeis.org
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
- 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).
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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
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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
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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
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LinearRecurrence[{3,2,5,-2,0,-1}, {1,2,7,29,109,416}, 30] (* G. C. Greubel, Oct 28 2019 *)
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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
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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
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