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A005036 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation and reflection.
(Formerly M1491)
14
1, 1, 2, 5, 16, 60, 261, 1243, 6257, 32721, 175760, 963900, 5374400, 30385256, 173837631, 1004867079, 5861610475, 34469014515, 204161960310, 1217145238485, 7299007647552, 44005602441840 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The subsequence of primes begins: 2, 5, 6257, no more through a(100). - Jonathan Vos Post, Apr 08 2011
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
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
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.
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.
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
KEYWORD
core,nonn,nice
AUTHOR
EXTENSIONS
More terms from Sascha Kurz, Oct 13 2001
Name edited by Andrew Howroyd, Nov 20 2017
STATUS
approved

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Last modified March 18 22:50 EDT 2024. Contains 370951 sequences. (Running on oeis4.)