A184785 - OEIS

%I #10 Mar 30 2012 18:37:25

%S 1,1,1,3,6,14,34,83,205,516,1317,3396,8848,23253,61570,164094,439860,

%T 1185086,3207477,8716726,23776459,65072379,178637758,491772915,

%U 1357288318,3754989329,10411112464,28924678247,80512118330,224501827180

%N Let A(x) satisfy: A(x) = 1 + x*A(x)^phi where phi = (sqrt(5)+1)/2, then this sequence equals the integer part of the coefficients of A(x).

%C Limit a(n+1)/a(n) = phi^sqrt(5) = phi^phi/(phi-1)^(phi-1) = 2.9329899...

%F a(n) = floor( binomial(phi*n, n)/(n/phi + 1) ) where phi = (sqrt(5)+1)/2.

%e G.f.: A(x) = 1 + x + c2*x^2 + c3*x^3 + c3*x^4 + c5*x^5 +...

%e A(x)^phi = 1 + c2*x + c3*x^2 + c4*x^3 + c5*x^4 + c6*x^5 +...

%e where the coefficients begin:

%e c2 = 1.6180339887..., c3 = 3.1180339887..., c4 = 6.5994579587...,

%e c5 = 14.818175608..., c6 = 34.657235589..., c7 = 83.517813823...,

%e c8 = 205.92186474..., c9 = 516.98843275..., c10 = 1317.122455..., ...;

%e the floor of the coefficients of A(x) forms this sequence.

%o (PARI) {a(n)=local(phi=(1+sqrt(5))/2);if(n<0,0,floor(binomial(phi*n,n)/((phi-1)*n+1)))}

%Y Cf. A184786.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jan 21 2011