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A000287 Number of rooted polyhedral graphs with n edges.
(Formerly M3290 N1326)
4
1, 0, 4, 6, 24, 66, 214, 676, 2209, 7296, 24460, 82926, 284068, 981882, 3421318, 12007554, 42416488, 150718770, 538421590, 1932856590, 6969847486, 25237057110, 91729488354, 334589415276, 1224445617889, 4494622119424 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,3

REFERENCES

A. J. W. Duijvestijn and P. J. Federico, The number of polyhedral (3-connected planar) graphs. Math. Comp. 37 (1981), no. 156, 523-532.

Handbook of Combinatorics, North-Holland '95, p. 892. (Gives different last term)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. T. Tutte, A new branch of enumerative graph theory, Bull. Amer. Math. Soc., 68 (1962), 500-504.

W. T. Tutte, A census of planar maps, Canad. J. Math., 15 (1963), 249-271.

Tutte, W. T. Three-connected planar maps. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 43--52. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0335323 (49 #105). - From N. J. A. Sloane, Jun 05 2012

Liu Yanpei, On the number of rooted c-nets, J. Combin. Theory, B 36 (1984), 118-123.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 6..1000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

FORMULA

a(n) = b(n-1) + 2*(-1)^n, n>=4, where b(3)=2, b(n) = [2*(2*n)!/(n!)^2 - (27*n^2+9*n-2)b(n-1)] / (54*n^2-90*n+32). [From Sean A. Irvine, Apr 14 2010]

(n + 4) a(n) = (3/2 n - 3) a(n - 1) + (8 n + 4) a(n - 2) + (15/2 n + 6) a(n - 3) + (2 n + 3) a(n - 4) [From Simon Plouffe, Feb 09 2012]

Liu Yanpei gives another recurrence. - N. J. A. Sloane, Mar 28 2012

a(n) ~ 2^(2*n+1)/(3^5*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Jul 19 2013

EXAMPLE

G.f. = x^6 + 4*x^8 + 6*x^9 + 24*x^10 + 66*x^11 + 214*x^12 + 676*x^13 + ...

MATHEMATICA

a[6] = 1; a[n_] := a[n] = ((9*(5 - 3*n)*n - 16)*a[n-1]*((n-1)!)^2 + 2*((-1)^n*(9*n*(3*n - 17) + 160)*((n-1)!)^2 + ((2*n - 2)!)))/(2*(9*n*(3*n - 11) + 88)*((n-1)!)^2); Table[ a[n], {n, 6, 31}] (* Jean-François Alcover, Oct 04 2011, after formula *)

CROSSREFS

Cf. A000256.

Sequence in context: A240290 A067001 A057343 * A032087 A165164 A241602

Adjacent sequences:  A000284 A000285 A000286 * A000288 A000289 A000290

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from Sean A. Irvine, Apr 14 2010.

Librandi b-file verified by N. J. A. Sloane, Mar 29 2012

STATUS

approved

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Last modified March 29 14:58 EDT 2017. Contains 284270 sequences.