A002709 Triangulations of the disk G_{n,0}.

1, 1, 1, 5, 24, 133, 846, 5661, 39556, 286000, 2123329, 16112057, 124512556, 977227830, 7772368380, 62535450861, 508271324688, 4168218286276, 34455941596060, 286864341314320, 2403705165816240, 20258850167232165, 171652324167433710, 1461462393790971585, 12498416291503945764

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Apparently, also the number of regular flexagons of order 3(n+1) (see Oakley-Wisner link pp. 149-151). - Michel Marcus, Jun 23 2013

References

• 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).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
• William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
• C. O. Oakley and R. J. Wisner, Flexagons, The American Mathematical Monthly, Vol. 64, No. 3 (Mar., 1957), pp. 143-154.

Crossrefs

Column k=0 of A262586.

Programs

PARI

a(n) = {if (n % 3 == 0, k = n/3; return (binomial(12*k-1,3*k-1)/((6*k-1)*(12*k-1)));); if (n % 3 == 1, k = (n-1)/3; return (binomial(12*k+3, 3*k)/(3*(4*k+1)*(6*k+1))+2*binomial(4*k,k)/(3*(3*k+1)));); if (n % 3 == 2, k = (n-2)/3; return (binomial(12*k+7,3*k+1)/(3*(2*k+1)*(12*k+7))+4*binomial(4*k+1,k)/(3*(3*k+2))););} \\ (number of regular flexagons of order 3*n) Michel Marcus, Jun 15 2013

Extensions

Extended by Max Alekseyev, Mar 30 2009

a(22) onwards from Andrew Howroyd, Nov 23 2024