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

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

Author

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

Keywords

• nonn
• easy
• nice

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]

Programs

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 *)

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

Extensions

More terms from Emeric Deutsch, Dec 19 2004