A005040 Number of nonequivalent dissections of a polygon into n pentagons by nonintersecting diagonals up to rotation and reflection. (Formerly M1851)

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

1,3

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 Christersson 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 Christersson, 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, 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, A survey of the cell-growth problem and some its variations, Com2Mac - Preprints.

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

G.f.: (8*G(z) - 3*G(z)^2 + 15*G(z^2) + 10z*G(z^2)^2 + 8z*G(z^5)) / 20, where G(z)=1+z*G(z)^4 is the g.f. for A002293. - Robert A. Russell, Oct 02 2025

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 *)

Crossrefs

Column k=5 of A295260.

Polyominoes: A005038 (oriented), A369471 (chiral), A369472 (achiral), A389194 (asymmetric), A002293 (rooted), A000207 {3,oo}, A005036 {4,oo}, A004127 {6,oo}, A005419 {7,oo}.

Sequence in context: A393726 A393727 A393728 * A380620 A191551 A263627

Adjacent sequences: A005037 A005038 A005039 * A005041 A005042 A005043

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sascha Kurz, Oct 13 2001

Name edited by Andrew Howroyd, Nov 20 2017

Status

approved