A005040 Number of nonequivalent dissections of a polygon into n pentagons by nonintersecting diagonals up to rotation and reflection.

1, 1, 2, 8, 33, 194, 1196, 8196, 58140, 427975, 3223610, 24780752, 193610550, 1534060440, 12302123640, 99699690472, 815521503060, 6725991120004, 55882668179880, 467387136083296, 3932600361607809, 33269692212847056, 282863689410850236, 2415930985594609548

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of unoriented polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,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 23 2024

References

• 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 = 1..1000
• 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, preprint, 2001.
• E. V. Konstantinova, Com2Mac - Preprints [Dead link?]

Crossrefs

Column k=5 of A295260.

Polyominoes: A005038 (oriented), A369471 (chiral), A369472 (achiral), A000207 {3,oo}, A005036 {4,oo}, A004127 {6,oo}, A005419 {7,oo}.

Programs

Mathematica

p=5; 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 *)

Formula

See Mathematica code.

a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Mar 13 2016

a(n) = A005038(n) - A369471(n) = (A005038(n) + A369472(n)) / 2 = A369471(n) + A369472(n). - Robert A. Russell, Jan 23 2024

Extensions

More terms from Sascha Kurz, Oct 13 2001.

Name edited by Andrew Howroyd, Nov 20 2017.