A002709 - OEIS

%I M3933 N1618 #36 Nov 23 2024 14:27:56

%S 1,1,1,5,24,133,846,5661,39556,286000,2123329,16112057,124512556,

%T 977227830,7772368380,62535450861,508271324688,4168218286276,

%U 34455941596060,286864341314320,2403705165816240,20258850167232165,171652324167433710,1461462393790971585,12498416291503945764

%N Triangulations of the disk G_{n,0}.

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

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

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

%H Andrew Howroyd, <a href="/A002709/b002709.txt">Table of n, a(n) for n = 0..500</a>

%H William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

%H William G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]

%H C. O. Oakley and R. J. Wisner, <a href="http://www.jstor.org/stable/2310544">Flexagons</a>, The American Mathematical Monthly, Vol. 64, No. 3 (Mar., 1957), pp. 143-154.

%o (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

%Y Column k=0 of A262586.

%K nonn

%O 0,4

%A _N. J. A. Sloane_

%E Extended by _Max Alekseyev_, Mar 30 2009

%E a(22) onwards from _Andrew Howroyd_, Nov 23 2024