OFFSET
0,3
COMMENTS
Limit a(n+1)/a(n) = phi^(phi+2) = (phi+1)^(phi+1)/phi^phi = 5.7032759...
FORMULA
a(n) = floor( binomial(phi^2*n, n)/(phi*n+1) ) where phi = (sqrt(5)+1)/2.
EXAMPLE
G.f.: A(x) = 1 + x + c2*x^2 + c3*x^3 + c3*x^4 + c5*x^5 +...
A(x)^(phi^2) = 1 + c2*x + c3*x^2 + c4*x^3 + c5*x^4 + c6*x^5 +...
where the coefficients begin:
c2 = 2.6180339887..., c3 = 8.9721359549..., c4 = 35.015865823...,
c5 = 147.58190992..., c6 = 654.49854850..., c7 = 3009.5978243...,
c8 = 14219.000049..., c9 = 68605.600329..., c10 = 336623.1131..., ...;
the floor of the coefficients of A(x) forms this sequence.
PROG
(PARI) {a(n)=local(phi=(1+sqrt(5))/2); if(n<0, 0, floor(binomial(phi^2*n, n)/(phi*n+1)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 21 2011
STATUS
approved