A006078 Number of triangulated (n+2)-gons rooted at an exterior edge. (Formerly M3822)

1, 1, 5, 12, 45, 143, 511, 1768, 6330, 22610, 81818, 297160, 1086813, 3991995, 14733435, 54587280, 203000094, 757398510, 2834519142, 10637507400, 40023665682, 150946230006, 570534682710, 2160865067312, 8199711750100

Offset

2,3

References

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

P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

Links

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]

Formula

Stockmeyer gives a g.f.

G.f.: (4*(1-x-x^2)-(1-2*x)(1-4*x)^(1/2)-3(1-4*x^2)^(1/2))/(8*x^2). - Emeric Deutsch, Dec 19 2004

a(n) ~ 2^(2*n-1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014

Maple

G:=(4*(1-x-x^2)-(1-2*x)*(1-4*x)^(1/2)-3*(1-4*x^2)^(1/2))/8/x^2: Gser:=series(G, x=0, 35): seq(coeff(Gser, x^n), n=2..28); # Emeric Deutsch, Dec 19 2004

Mathematica

g:=(4*(1-x-x^2)-(1-2*x)*(1-4*x)^(1/2)-3*(1-4*x^2)^(1/2))/8/x^2; gser := Series[g, {x, 0, 26}]; Drop[ CoefficientList[gser, x], 2] (* Jean-François Alcover, Apr 06 2012, after Emeric Deutsch *)
Drop[CoefficientList[Series[(4(1-x-x^2)- (1-2x)Sqrt[1-4x]- 3Sqrt[1- 4x^2])/(8x^2), {x, 0, 30}], x], 2] (* Harvey P. Dale, Apr 07 2013 *)

Crossrefs

Sequence in context: A052644 A052280 A195541 * A046612 A254217 A211951

Adjacent sequences: A006075 A006076 A006077 * A006079 A006080 A006081

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

Extensions

More terms from Emeric Deutsch, Dec 19 2004

Status

approved