A184785 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).

1, 1, 1, 3, 6, 14, 34, 83, 205, 516, 1317, 3396, 8848, 23253, 61570, 164094, 439860, 1185086, 3207477, 8716726, 23776459, 65072379, 178637758, 491772915, 1357288318, 3754989329, 10411112464, 28924678247, 80512118330, 224501827180

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..29.

Formula

a(n) = floor( binomial(phi*n, n)/(n/phi + 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 = 1 + c2*x + c3*x^2 + c4*x^3 + c5*x^4 + c6*x^5 +...
where the coefficients begin:
c2 = 1.6180339887..., c3 = 3.1180339887..., c4 = 6.5994579587...,
c5 = 14.818175608..., c6 = 34.657235589..., c7 = 83.517813823...,
c8 = 205.92186474..., c9 = 516.98843275..., c10 = 1317.122455..., ...;
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*n, n)/((phi-1)*n+1)))}

Crossrefs

Cf. A184786.

Sequence in context: A078062 A275873 A018017 * A350678 A332362 A196479

Adjacent sequences: A184782 A184783 A184784 * A184786 A184787 A184788

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2011

Status

approved