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A033505
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Expansion of 1/(1 - 3*x - x^2 + x^3).
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11
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1, 3, 10, 32, 103, 331, 1064, 3420, 10993, 35335, 113578, 365076, 1173471, 3771911, 12124128, 38970824, 125264689, 402640763, 1294216154, 4160024536, 13371648999, 42980755379, 138153890600, 444070778180, 1427385469761, 4588073296863, 14747534582170
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of ways to tile, with squares and dominoes, a 2 X n board with one extra space at the end. Here is the board for n=3:
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and here is one of the a(3)=32 possible tilings of this board:
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(End)
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LINKS
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FORMULA
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a(n) = 3*a(n-1) + a(n-2) - a(n-3). - Greg Dresden, Aug 16 2018
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MAPLE
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seq(coeff(series(1/(1-3*x-x^2+x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 14 2019
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MATHEMATICA
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CoefficientList[Series[1/(1-3x-x^2+x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 1, -1}, {1, 3, 10}, 30] (* Vincenzo Librandi, Aug 17 2018 *)
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PROG
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(Magma) I:=[1, 3, 10]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 17 2018
(PARI) my(x='x+O('x^30)); Vec(1/(1-3*x-x^2+x^3)) \\ G. C. Greubel, Oct 14 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/(1-3*x-x^2+x^3)).list()
(GAP) a:=[1, 3, 10];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Oct 14 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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