A005034 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation.

1, 1, 1, 2, 7, 25, 108, 492, 2431, 12371, 65169, 350792, 1926372, 10744924, 60762760, 347653944, 2009690895, 11723100775, 68937782355, 408323229930, 2434289046255, 14598011263089, 88011196469040, 533216750567280, 3245004785069892, 19829768942544276, 121639211516546668

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also, with a different offset, number of colored quivers in the 2-mutation class of a quiver of Dynkin type A_n. - N. J. A. Sloane, Jan 22 2013

Closed formula is given in my paper linked below. - Nikos Apostolakis, Aug 01 2018

Number of oriented polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,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 20 2024

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Nikos Apostolakis, Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections, arXiv:1807.11602 [math.CO], July 2018.
• Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
• F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
• F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Quebec 16 (1992), no 1, 53-80.
• P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
• Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233. See p. 232.
• Torkildsen, Hermund A., Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133.

Crossrefs

Column k=4 of A295224.

Polyominoes: A005036 (unoriented), A369315 (chiral), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A001683(n+2) {3,oo}, A005038 {5,oo}.

Programs

Mathematica

p=4; 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)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1}]], {n, 0, 20}] (* Robert A. Russell, Dec 11 2004 *)
Table[(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1)-If[OddQ[n],-2Binomial[(3n-1)/2,(n-1)/2]-If[1==Mod[n,4],4Binomial[(3n-3)/4,(n-1)/4],0],-2Binomial[3n/2,n/2]]/(n+1))/4,{n,0,30}] (* Robert A. Russell, Jun 19 2025 *)

Formula

a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Mar 13 2016

a(n) = A005036(n) + A369315(n) = 2*A005036(n) - A047749(n) = 2*A369315(n) + A047749(n). - Robert A. Russell, Jan 19 2024

G.f.: (3*G(z) - G(z)^2 + 2*G(z^2) + z*G(z^2)^2 + 2z*G(z^4)) / 4, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Jun 19 2025

Extensions

Name clarified by Andrew Howroyd, Nov 20 2017