A005036 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation and reflection. (Formerly M1491)

1, 1, 2, 5, 16, 60, 261, 1243, 6257, 32721, 175760, 963900, 5374400, 30385256, 173837631, 1004867079, 5861610475, 34469014515, 204161960310, 1217145238485, 7299007647552, 44005602441840

Offset

1,3

Comments

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

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

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..100

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.

E. V. Konstantinova, A survey of the cell-growth problem and some its variations, Com 2 MaC-KOSEF, 2001.

Index entries for "core" sequences

Formula

a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 4)). - Vaclav Kotesovec, Mar 13 2016

a(n) = A005034(n) - A369315(n) = (A005034(n) + A047749(n)) / 2 = A369315(n) + A047749(n). - Robert A. Russell, Jan 19 2024

Mathematica

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 *)

Crossrefs

Column k=4 of A295260.

Cf. A005419, A004127.

Polyominoes: A005034 (oriented), A369315 (chiral), A047749 (achiral), A000207 {3,oo}, A005040 {5,oo}, A005038 {5,oo} (oriented).

Sequence in context: A205486 A210668 A279564 * A012051 A012159 A009736

Adjacent sequences: A005033 A005034 A005035 * A005037 A005038 A005039

Keyword

core,nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Sascha Kurz, Oct 13 2001

Name edited by Andrew Howroyd, Nov 20 2017

Status

approved