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%I #32 Jan 26 2024 08:36:02
%S 1,1,3,19,118,931,7756,68685,630465,5966610,57805410,571178751,
%T 5737638778,58455577800,602859152496,6283968796705,66119469155523,
%U 701526880303315,7498841128986109,80696081185766970,873654669882574580
%N Number of colored quivers in the 4-mutation class of a quiver of Dynkin type A_n.
%C Also, number of nonequivalent dissections of a polygon into n+1 hexagons by nonintersecting diagonals up to rotation. - _Andrew Howroyd_, Nov 20 2017
%C 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
%H Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H Hermund A. Torkildsen, <a href="https://arxiv.org/abs/1004.4512">Colored quivers of type A and the cell-growth problem</a>, arXiv:1004.4512 [math.RT], 2010.
%H Hermund A. Torkildsen, <a href="http://dx.doi.org/10.1142/S0219498812501332">Colored quivers of type A and the cell-growth problem</a>, J. Algebra and Applications, 12 (2013), #1250133.
%F a(n) ~ 5^(5*n + 11/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 27/2)). - _Vaclav Kotesovec_, Jun 15 2018
%F a(n-1) = A004127(n) + A369473(n) = 2*A004127(n) - A143546(n) = 2*A369473(n) + A143546(n). - _Robert A. Russell_, Jan 23 2024
%t u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
%t 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;
%t a[n_] := T[n + 1, 6];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd_ *)
%t 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 *)
%Y Column k=6 of A295224.
%Y Polyominoes: A004127 (unoriented), A369473 (chiral), A143546 (achiral), A001683(n+2) {3,oo}, A005034 {4,oo}, A005038 {5,oo}.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Jan 22 2013
%E a(0)=1 and a(18)-a(20) corrected by _Andrew Howroyd_, Nov 20 2017
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