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A004127 Number of planar hexagon trees with n hexagons.
(Formerly M2936)
9
1, 1, 3, 12, 68, 483, 3946, 34485, 315810, 2984570, 28907970, 285601251, 2868869733, 29227904840, 301430074416, 3141985563575, 33059739636198, 350763452126835, 3749420616902637, 40348040718155170, 436827335493148600 (list; graph; refs; listen; history; text; internal format)
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

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]

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

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. A005419, A005040, A002294.

Sequence in context: A121812 A039750 A296979 * 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

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Last modified August 17 03:44 EDT 2022. Contains 356181 sequences. (Running on oeis4.)