cp's OEIS Frontend

A005036 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation and reflection.

%I A005036 M1491 #77 Jun 22 2025 00:17:35
%S A005036 1,1,2,5,16,60,261,1243,6257,32721,175760,963900,5374400,30385256,
%T A005036 173837631,1004867079,5861610475,34469014515,204161960310,
%U A005036 1217145238485,7299007647552,44005602441840
%N A005036 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation and reflection.
%C A005036 Closed formula is given in my paper linked below. - _Nikos Apostolakis_, Aug 01 2018
%C A005036 Number of unoriented 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 unoriented polyominoes, chiral pairs are counted as one. - _Robert A. Russell_, Jan 20 2024
%D A005036 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005036 T. D. Noe, <a href="/A005036/b005036.txt">Table of n, a(n) for n = 1..100</a>
%H A005036 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 A005036 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A005036 F. Harary, E. M. Palmer, and R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>
%H A005036 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 A005036 E. V. Konstantinova, <a href="https://web.archive.org/web/20040225165049/http://com2mac.postech.ac.kr/papers/2001/01-06.pdf">A survey of the cell-growth problem and some its variations</a>, Com 2 MaC-KOSEF, 2001. (Archived link.)
%H A005036 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A005036 a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 4)). - _Vaclav Kotesovec_, Mar 13 2016
%F A005036 a(n) = A005034(n) - A369315(n) = (A005034(n) + A047749(n)) / 2 = A369315(n) + A047749(n). - _Robert A. Russell_, Jan 19 2024
%F A005036 G.f.: (3*G(z) - G(z)^2 + 6*G(z^2) + 5z*G(z^2)^2 + 2z*G(z^4)) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - _Robert A. Russell_, Jun 19 2025
%t A005036 p=4; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(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, 2}]])/2, {n, 1, 20}] (* _Robert A. Russell_, Dec 11 2004 *)
%t A005036 Table[(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1)-If[OddQ[n],-10Binomial[(3n-1)/2,(n-1)/2]-If[1==Mod[n,4],4Binomial[(3n-3)/4,(n-1)/4],0],-6Binomial[3n/2,n/2]]/(n+1))/8,{n,0,30}] (* _Robert A. Russell_, Jun 19 2025 *)
%Y A005036 Column k=4 of A295260.
%Y A005036 Cf. A005419, A004127.
%Y A005036 Polyominoes: A005034 (oriented), A369315 (chiral), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A000207 {3,oo}, A005040 {5,oo}, A005038 {5,oo} (oriented).
%K A005036 core,nonn,nice
%O A005036 1,3
%A A005036 _N. J. A. Sloane_
%E A005036 More terms from _Sascha Kurz_, Oct 13 2001
%E A005036 Name edited by _Andrew Howroyd_, Nov 20 2017