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A221184
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Number of colored quivers in the 4-mutation class of a quiver of Dynkin type A_n.
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6
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1, 1, 3, 19, 118, 931, 7756, 68685, 630465, 5966610, 57805410, 571178751, 5737638778, 58455577800, 602859152496, 6283968796705, 66119469155523, 701526880303315, 7498841128986109, 80696081185766970, 873654669882574580
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OFFSET
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0,3
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COMMENTS
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Also, number of nonequivalent dissections of a polygon into n+1 hexagons by nonintersecting diagonals up to rotation. - Andrew Howroyd, Nov 20 2017
Number of oriented polyominoes composed of n+1 hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 23 2024
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LINKS
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FORMULA
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a(n) ~ 5^(5*n + 11/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 27/2)). - Vaclav Kotesovec, Jun 15 2018
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MATHEMATICA
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u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := u[n, k, 1] + (If[EvenQ[n], u[n/2, k, 1], 0] - u[n, k, 2])/2 + DivisorSum[GCD[n - 1, k], EulerPhi[#]*u[(n - 1)/#, k, k/#] &]/k;
a[n_] := T[n + 1, 6];
p=6; Table[Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2))+If[OddQ[n], 0, Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+DivisorSum[GCD[p, n-1], EulerPhi[#]Binomial[((p-1)n+1)/#, (n-1)/#]/((p-1)n+1)&, #>1&], {n, 30}] {* Robert A. Russell, Jan 23 2024 *)
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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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EXTENSIONS
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STATUS
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approved
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