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A357771
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Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-6 node.
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1
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1, 6, 60, 744, 10224, 148896, 2250816, 34922880, 552386304, 8867756544, 144044098560, 2362292213760, 39049785446400, 649843233546240, 10876273137008640, 182934715370471424, 3090181365862170624, 52398620697685524480, 891492911924665122816, 15213249205591283859456, 260315328935885892747264
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OFFSET
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0,2
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COMMENTS
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Paths that return to the same point in a quasi-regular rhombic lattice must always have even length (i.e., 2n) because of parity: degree-6 nodes alternate with degree-3 nodes.
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (binomial(n, k) * Sum_{j=0..n} (binomial(n, j) * Sum_{i= 0..j} ((1/(2^j))*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))). - Detlef Meya, May 15 2024
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EXAMPLE
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a(2)=60, because there are 6*6=36 paths that visit one of six adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; plus 6*4=24 paths that pass through one of the six vertices at distance 2, leaving and returning via any of two available paths to that vertex; all resulting in a closed path of length 2n=2*2=4.
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MATHEMATICA
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a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^j))*Binomial[2*i, j]*Binomial[2*i, i]*Binomial[2*(j-i), j-i], {i, 0, j}], {j, 0, n}], {k, 0, n}]; Flatten[Table[a[n], {n, 0, 17}]] (* Detlef Meya, May 15 2024 *)
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n, k) * sum(j=0, n, binomial(n, j) * sum(i=0, j, (1/(2^j)*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))))); \\ Michel Marcus, May 20 2024
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CROSSREFS
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The accompanying sequences for the number of paths that return to a degree-3 node is A357770.
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KEYWORD
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nonn,easy,walk,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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