This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A003409 M2814 #43 Aug 31 2025 23:43:40
%S A003409 3,9,30,105,378,1386,5148,19305,72930,277134,1058148,4056234,15600900,
%T A003409 60174900,232676280,901620585,3500409330,13612702950,53017895700,
%U A003409 206769793230,807386811660,3156148445580,12350146091400,48371405524650,189615909656628,743877799422156
%N A003409 a(n) = 3*binomial(2n-1,n).
%C A003409 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
%D A003409 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003409 T. D. Noe, <a href="/A003409/b003409.txt">Table of n, a(n) for n = 1..200</a>
%H A003409 C. Domb and A. J. Barrett, <a href="http://dx.doi.org/10.1016/0012-365X(74)90081-8">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358.
%H A003409 C. Domb & A. J. Barrett, <a href="/A003408/a003408.pdf">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)
%H A003409 C. Domb & A. J. Barrett, <a href="/A001764/a001764.pdf">Notes on Table 2 in "Enumeration of ladder graphs"</a>, Discrete Math. 9 (1974), 55. (Annotated scanned copy)
%F A003409 a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015
%F A003409 From _Stefano Spezia_, Jul 05 2021: (Start)
%F A003409 O.g.f.: 6*x/((1 - sqrt(1 - 4*x))*sqrt(1 - 4*x)) - 3.
%F A003409 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.
%F A003409 a(n) ~ 3*4^n/(2*sqrt(n*Pi)). (End)
%F A003409 D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jul 22 2025
%p A003409 a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n)):
%p A003409 seq(a(n), n=1..26); # _Peter Luschny_, Dec 14 2015
%t A003409 Table[3*Binomial[2*n - 1, n], {n, 20}] (* _T. D. Noe_, Oct 07 2013 *)
%o A003409 (PARI) a(n) = 3*binomial(2*n-1,n) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A003409 Equals 3 * A001700.
%K A003409 nonn,changed
%O A003409 1,1
%A A003409 _N. J. A. Sloane_
%E A003409 More terms from _Jon E. Schoenfield_, Mar 26 2010