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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] 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 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.)