A005038 Number of nonequivalent dissections of a polygon into n pentagons by nonintersecting diagonals up to rotation. (Formerly M2026)

1, 1, 2, 12, 57, 366, 2340, 16252, 115940, 854981, 6444826, 49554420, 387203390, 3068067060, 24604111560, 199398960212, 1631041938108, 13451978877748, 111765327780200, 934774244822704, 7865200653146905

Offset

1,3

Comments

Also, with a different offset, number of colored quivers in the 3-mutation class of a quiver of Dynkin type A_n. - N. J. A. Sloane, Jan 22 2013

Number of oriented 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 oriented polyominoes, chiral pairs are counted as two. - 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.

Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, arXiv:1004.4512 [math.RT], 2010.

Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133. - From N. J. A. Sloane, Jan 22 2013

Formula

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

a(n) = A005040(n) + A369471(n) = 2*A005040(n) - A369472(n) = 2*A369471(n) + A369472(n). - Robert A. Russell, Jan 23 2024

Mathematica

p=5; Table[Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) +If[OddQ[n], 0, Binomial[(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}]], {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)

Crossrefs

Column k=5 of A295224.

Polyominoes: A005040 (unoriented), A369471 (chiral), A369472 (achiral), A001683(n+2) {3,oo}, A005034 {4,oo}, A221184{n-1} {6,oo}.

Sequence in context: A363402 A067125 A177782 * A094780 A268594 A100103

Adjacent sequences: A005035 A005036 A005037 * A005039 A005040 A005041

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

a(21) corrected by Sean A. Irvine, Mar 11 2016

Name edited by Andrew Howroyd, Nov 20 2017

Status

approved