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A338672
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Number of n-step closed walks on the kagomé lattice.
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0
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1, 0, 4, 4, 28, 60, 264, 784, 3004, 10204, 37824, 135784, 502784, 1851200, 6901696, 25766144, 96797244, 364655100, 1379120400, 5230011896, 19890313128, 75823622984, 289698620336, 1109059301536, 4253731156128, 16342545417760, 62885474132992, 242331022479040, 935085717105792, 3612737418620032, 13974224404904704
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) is the constant coefficient in the expansion of 1/3 * trace(A^n), where A is the matrix {{0, y+x*y, y+1/x}, {1/y+1/(x*y), 0, 1/x+1/(x*y)}, {x+1/y, x+x*y, 0}}.
a(n) = (1/3) * ((-2)^n + 2 * Sum_{k=0..floor(n/2)} Sum_{j=0..k} binomial(n, 2*k) * binomial(k, j)^2 * binomial(2*j, j)).
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MATHEMATICA
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a[n_] := ((-2)^n + 2 Sum[Binomial[n, 2 k] Binomial[k, j]^2 Binomial[2 j, j], {k, 0, Floor[n/2]}, {j, 0, k}])/3; Table[a[n], {n, 0, 30}]
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PROG
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(PARI) a(n)={((-2)^n + 2 * sum(k=0, n\2, sum(j=0, k, binomial(n, 2*k) * binomial(k, j)^2 * binomial(2*j, j))))/3} \\ Andrew Howroyd, Apr 24 2021
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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Li GAN and Stéphane Ouvry, Apr 23 2021
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STATUS
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approved
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