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A123520
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Number of vertical dominoes in all possible tilings of a 2n X 3 grid by dominoes.
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2
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4, 28, 152, 744, 3436, 15284, 66224, 281424, 1178196, 4874444, 19973192, 81189688, 327817404, 1316035940, 5257118560, 20909651104, 82849544868, 327163551612, 1288036695544, 5057236343176, 19807689093644, 77408388584724
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} 2^(k+1) * k * C(n+k,2*k).
a(n) = Sum_{k=0..n} k * A123519(n,k).
G.f.: 4*z*(1-z)/(1-4*z+z^2)^2.
a(n) = (2+sqrt(3))^n*((1+sqrt(3))*n+1/sqrt(3))/3 + (2-sqrt(3))^n*((1-sqrt(3))*n-1/sqrt(3))/3. - Vaclav Kotesovec, Nov 29 2012
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EXAMPLE
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a(1) = 4 because a 2 X 3 grid can be tiled in 3 ways with dominoes: 3 horizontal dominoes, 1 horizontal domino above two adjacent vertical dominoes and 1 horizontal domino below two adjacent vertical dominoes; these have altogether 4 vertical dominoes.
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MAPLE
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a:=n->sum(k*2^(k+1)*binomial(n+k, 2*k), k=0..n): seq(a(n), n=1..24);
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MATHEMATICA
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FullSimplify[Table[(2+Sqrt[3])^n*((1+Sqrt[3])*n+1/Sqrt[3])/3 + (2-Sqrt[3])^n*((1-Sqrt[3])*n-1/Sqrt[3])/3, {n, 1, 20}]] (* Vaclav Kotesovec, Nov 29 2012 *)
Table[Sum[2^(k + 1)*k*Binomial[n + k, 2 k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 14 2017 *)
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PROG
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(PARI) z='z+O('z^50); Vec(4*z*(1-z)/(1-4*z+z^2)^2) \\ G. C. Greubel, Oct 14 2017
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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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