A004127 Number of planar hexagon trees with n hexagons. (Formerly M2936)

1, 1, 3, 12, 68, 483, 3946, 34485, 315810, 2984570, 28907970, 285601251, 2868869733, 29227904840, 301430074416, 3141985563575, 33059739636198, 350763452126835, 3749420616902637, 40348040718155170, 436827335493148600

Offset

1,3

Comments

Number of nonequivalent dissections of a polygon into n hexagons by nonintersecting diagonals up to rotation and reflection. - Andrew Howroyd, Nov 20 2017

Number of unoriented polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,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..925

L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147.

L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147. [Annotated scanned copy]

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

Index entries for sequences related to trees

Formula

See Theorem 3 on p. 142 in the Beineke-Pippert paper; also the Maple and Mathematica codes here.

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

a(n) = A221184(n-1) - A369473(n) = (A221184(n-1) + A143546(n)) / 2 = A369473(n) + A143546(n). - Robert A. Russell, Jan 23 2024

G.f.: (5*G(z) - 2*G(z)^2 + 9*G(z^2) + 7*z*G(z^2)^3 + 2*z*G(z^3)^2 + 2*z*G(z^6)) / 12, where G(z)=1+z*G(z)^5 is the g.f. for A002294. - Robert A. Russell, Oct 03 2025

Maple

T := proc(n) if floor(n)=n then binomial(5*n+1, n)/(5*n+1) else 0 fi end: U := proc(n) if n mod 2 = 0 then binomial(5*n/2+1, n/2)/(5*n/2+1) else 6*binomial((5*n+1)/2, (n-1)/2)/(5*n+1) fi end: S := n->T(n)/4/(2*n+1)+T(n/2)/6+(5*n-2)*T((n-1)/3)/6/(2*n+1)+T((n-1)/6)/6+7*U(n)/12: seq(S(n), n=1..25); # Emeric Deutsch

Mathematica

p=6; 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=6 of A295260.

Cf. A002294.

Polyominoes: A221184{n-1} (oriented), A369473 (chiral), A143546 (achiral), A389337 (asymmetric), A002294 (rooted), A005040 {5,oo}, A005419 {7,oo}.

Sequence in context: A393731 A393732 A393733 * A058115 A101313 A257605

Adjacent sequences: A004124 A004125 A004126 * A004128 A004129 A004130

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Jan 22 2004

Status

approved