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A357770
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Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-3 node.
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1
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1, 3, 30, 372, 5112, 74448, 1125408, 17461440, 276193152, 4433878272, 72022049280, 1181146106880, 19524892723200, 324921616773120, 5438136568504320, 91467357685235712, 1545090682931085312, 26199310348842762240, 445746455962332561408, 7606624602795641929728
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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-3 nodes alternate with degree-6 nodes.
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LINKS
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FORMULA
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a(0) = 1; a(n) = Sum_{k=0..n} (binomial(n, k) * Sum_{j=0..n} (binomial(n, j) * Sum_{i= 0..j} ((1/(2^(j + 1)))*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))). - Detlef Meya, May 20 2024
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EXAMPLE
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a(2)=30, because there are 3*3=9 paths that visit one of three adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; 3*5=15 paths visiting one of five distance-2 vertices that are adjacent to the three adjacent vertices; plus 3*2=6 paths traversing the perimeter of three adjacent rhombi in counterclockwise or clockwise direction; all resulting in a closed path of length 2n=2*2=4.
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MATHEMATICA
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a[0] := 1; a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^(j+1)))*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, 19}]] (* Detlef Meya, 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-6 node is A357771.
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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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