A005419 Number of nonequivalent dissections of a polygon into n heptagons by nonintersecting diagonals up to rotation and reflection. (Formerly M3023)

1, 1, 3, 16, 112, 1020, 10222, 109947, 1230840, 14218671, 168256840, 2031152928, 24931793768, 310420597116, 3912823963482, 49853370677834, 641218583442360, 8316918403772790, 108686334145327785, 1429927553582849256, 18927697628428129728, 251931892228273729375

Offset

1,3

Comments

Number of unoriented polyominoes with n heptagonal cells of the hyperbolic regular tiling with Schläfli symbol {7,oo}. A stereographic projection of the {7,oo} tiling on the Poincaré disk can be obtained via the Christersson link. For unoriented polyominoes, chiral pairs are counted as one. - Robert A. Russell, Oct 21 2025

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..850

Malin Christersson, Make hyperbolic tilings of images, web page, 2019.

Frank Harary, Edgar M. Palmer and Ronald C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

Formula

See Mathematica code.

a(n) ~ 2^(6*n - 1) * 3^(6*n + 1/2) / (sqrt(Pi) * n^(5/2) * 5^(5*n + 5/2)). - Vaclav Kotesovec, Mar 13 2016

From Robert A. Russell, Oct 21 2025: (Start)

G.f.: (12*G(z) - 5*G(z)^2 + 21*G(z^2) + 14*z*G(z^2)^3 + 12*z*G(z^7)) / 28, where G(z) = 1 + z*G(z)^6 is the g.f. for A002295.

a(n) = A389561(n) - A389562(n) = (A389561(n) + A389563(n)) / 2 = A389562(n) + A389563(n). (End)

Mathematica

p=7; 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=7 of A295260.

Cf. A389561 (oriented), A389562 (chiral), A389563 (achiral), A389564 (asymmetric), A002295 (rooted), A004127 {6,oo}, A389937 {8,oo}.

Sequence in context: A393736 A393737 A393738 * A379193 A390715 A381029

Adjacent sequences: A005416 A005417 A005418 * A005420 A005421 A005422

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Robert A. Russell, Dec 11 2004

Name edited by Andrew Howroyd, Nov 20 2017

Status

approved