cp's OEIS Frontend

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

%I A005034 M1768 #69 Jun 20 2025 10:46:34
%S A005034 1,1,1,2,7,25,108,492,2431,12371,65169,350792,1926372,10744924,
%T A005034 60762760,347653944,2009690895,11723100775,68937782355,408323229930,
%U A005034 2434289046255,14598011263089,88011196469040,533216750567280,3245004785069892,19829768942544276,121639211516546668
%N A005034 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation.
%C A005034 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
%C A005034 Closed formula is given in my paper linked below. - _Nikos Apostolakis_, Aug 01 2018
%C A005034 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
%D A005034 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
%D A005034 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005034 G. C. Greubel, <a href="/A005034/b005034.txt">Table of n, a(n) for n = 0..1000</a>
%H A005034 Nikos Apostolakis, <a href="https://arxiv.org/abs/1807.11602">Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections</a>, arXiv:1807.11602 [math.CO], July 2018.
%H A005034 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A005034 F. Harary, E. M. Palmer, R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>
%H A005034 F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389.
%H A005034 P. Leroux and B. Miloudi, <a href="http://www.labmath.uqam.ca/~annales/volumes/16-1/PDF/053-080.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Quebec 16 (1992), no 1, 53-80.
%H A005034 P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H A005034 Alexander Stoimenow, <a href="https://doi.org/10.1016/S0012-365X(99)00347-7">On the number of chord diagrams</a>, Discr. Math. 218 (2000), 209-233. See p. 232.
%H A005034 Torkildsen, Hermund A., <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 A005034 a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 3)). - _Vaclav Kotesovec_, Mar 13 2016
%F A005034 a(n) = A005036(n) + A369315(n) = 2*A005036(n) - A047749(n) = 2*A369315(n) + A047749(n). - _Robert A. Russell_, Jan 19 2024
%F A005034 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
%t A005034 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 *)
%t A005034 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 *)
%Y A005034 Column k=4 of A295224.
%Y A005034 Polyominoes: A005036 (unoriented), A369315 (chiral), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A001683(n+2) {3,oo}, A005038 {5,oo}.
%K A005034 nonn,easy
%O A005034 0,4
%A A005034 _N. J. A. Sloane_
%E A005034 Name clarified by _Andrew Howroyd_, Nov 20 2017