OFFSET
0,3
COMMENTS
Limit a(n+1)/a(n) = (sqrt(5)+3)/2.
FORMULA
a(n) = (A051286(n) + A000045(n+1))/2, where A000045(n+1) = Fibonacci(n+1) and A051286(n) = Whitney number of level n.
G.f.: ( 1/(1-x-x^2) + 1/sqrt( (1+x+x^2)*(1-3*x+x^2) ) )/2. - Michael Somos, Jul 27 2007
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} x^k * C(n,k)*(C(n,k) + 1)/2. - Paul D. Hanna, Aug 13 2014
PROG
(PARI) a(n)=(sum(k=0, n, binomial(n-k, k)^2)+fibonacci(n+1))/2
(PARI) {a(n)= if(n<0, 0, polcoeff( (1/(1-x-x^2) +1/sqrt((1+x+x^2)* (1-3*x+x^2)+ x*O(x^n)))/2, n))} /* Michael Somos, Jul 27 2007 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 22 2005
STATUS
approved