OFFSET
0,2
COMMENTS
Or, start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central triangle containing exactly 3 numbers. The first three triangles are
...1...........2.........6
.1...1.......3...3.....10..10
and the corresponding sums are 3, 8 and 26. - Amarnath Murthy, Mar 25 2003
This sequence starting at a(n+2) has Hankel transform A000032(2n+1)*2^n (empirical observation). - Tony Foster III, May 20 2016
FORMULA
G.f.: (x+1)/sqrt(1-4*x). - Vladeta Jovovic, Jan 08 2004
a(n) = 2binomial(2n-1, n-1)+binomial(2n-2, n-1). - Emeric Deutsch, Apr 20 2004
a(n) = (n+1)*C(n) + n*C(n-1), C = Catalan number (A000108). - Gary W. Adamson, Dec 28 2007
G.f.: G(0) where G(k)= 1 + x/(1 - (4*k+2)/((4*k+2) + (k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012
D-finite with recurrence n*a(n) -3*n*a(n-1) +2*(-2*n+5)*a(n-2)=0. - R. J. Mathar, May 01 2024
MAPLE
seq(binomial(2*n, n)+binomial(2*n-2, n-1), n=0..24);
seq(2*binomial(2*n-1, n-1)+binomial(2*n-2, n-1), n=1..24);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers
STATUS
approved