A002709 Triangulations of the disk G_{n,0}. (Formerly M3933 N1618)

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

0,4

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.

Prog

(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

Crossrefs

Column k=0 of A262586.

Sequence in context: A036919 A020067 A066118 * A193668 A009411 A080996

Adjacent sequences: A002706 A002707 A002708 * A002710 A002711 A002712

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended by Max Alekseyev, Mar 30 2009

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

Status

approved