A003409 a(n) = 3*binomial(2n-1,n).

3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, 206769793230, 807386811660, 3156148445580, 12350146091400, 48371405524650, 189615909656628, 743877799422156

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• changed

Comments

a(n) is the number of ways to tile a bracelet of length 3*n with n squares and n dominos, or in other words, the number of ways to have exactly n pairings on the cycle graph C_(3*n). - Greg Dresden and Zhengyu Zhang, Aug 15 2025

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..200
• C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
• C. Domb & A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)
• C. Domb & A. J. Barrett, Notes on Table 2 in "Enumeration of ladder graphs", Discrete Math. 9 (1974), 55. (Annotated scanned copy)

Crossrefs

Equals 3 * A001700.

Programs

Maple

a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(a(n), n=1..26); # Peter Luschny, Dec 14 2015

Mathematica

Table[3*Binomial[2*n - 1, n], {n, 20}] (* T. D. Noe, Oct 07 2013 *)

PARI

a(n) = 3*binomial(2*n-1,n) \\ Charles R Greathouse IV, Oct 23 2023

Formula

a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015

From Stefano Spezia, Jul 05 2021: (Start)

O.g.f.: 6*x/((1 - sqrt(1 - 4*x))*sqrt(1 - 4*x)) - 3.

E.g.f.: 3*(exp(2*x)*I_0(2*x) - 1)/2, where I_n(x) is the modified Bessel function of the first kind.

a(n) ~ 3*4^n/(2*sqrt(n*Pi)). (End)

D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jul 22 2025

Extensions

More terms from Jon E. Schoenfield, Mar 26 2010