A221184 Number of colored quivers in the 4-mutation class of a quiver of Dynkin type A_n.

1, 1, 3, 19, 118, 931, 7756, 68685, 630465, 5966610, 57805410, 571178751, 5737638778, 58455577800, 602859152496, 6283968796705, 66119469155523, 701526880303315, 7498841128986109, 80696081185766970, 873654669882574580

Offset: 0

Author

N. J. A. Sloane, Jan 22 2013

Keywords

• nonn
• easy

Comments

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

Links

• Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
• Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, arXiv:1004.4512 [math.RT], 2010.
• Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133.

Crossrefs

Column k=6 of A295224.

Polyominoes: A004127 (unoriented), A369473 (chiral), A143546 (achiral), A001683(n+2) {3,oo}, A005034 {4,oo}, A005038 {5,oo}.

Programs

Mathematica

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];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)
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 *)

Formula

a(n) ~ 5^(5*n + 11/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 27/2)). - Vaclav Kotesovec, Jun 15 2018

a(n-1) = A004127(n) + A369473(n) = 2*A004127(n) - A143546(n) = 2*A369473(n) + A143546(n). - Robert A. Russell, Jan 23 2024

Extensions

a(0)=1 and a(18)-a(20) corrected by Andrew Howroyd, Nov 20 2017