A186577 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^[n*phi^2] where phi = (1+sqrt(5))/2.

1, 1, 3, 13, 65, 353, 2025, 12074, 74083, 464708, 2966647, 19211268, 125890754, 833242554, 5562338802, 37406660537, 253185542824, 1723428340232, 11790556480270, 81026845851661, 559086953351167, 3871831119088196

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f. satisfies: A(x) = F(x*A(x)) where A(x/F(x)) = F(x) is the g.f. of A186576, which in turn satisfies: F(x) = Sum_{n>=0} x^n*F(x)^[n*phi].

G.f.: A(x) = (1/x)*Series_Reversion(x/F(x)) where F(x) is the g.f. of A186576.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 65*x^4 + 353*x^5 + 2025*x^6 +...
The g.f. satisfies:
A(x) = 1 + x*A(x)^2 + x^2*A(x)^5 + x^3*A(x)^7 + x^4*A(x)^10 + x^5*A(x)^13 + x^6*A(x)^15 + x^7*A(x)^18 +...+ x^n*A(x)^A001950(n) +...
The g.f. of A186576, F(x) = A(x/F(x)), satisfies:
F(x) = 1 + x*F(x) + x^2*F(x)^3 + x^3*F(x)^4 + x^4*F(x)^6 + x^5*F(x)^8 + x^6*F(x)^9 + x^7*F(x)^11 +...+ x^n*F(x)^A000201(n) +...
and begins:
F(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 72*x^5 + 274*x^6 +...
Since A(x) = F(x*A(x)), then:
A(x) = 1 + x*A(x) + 2*x^2*A(x)^2 + 6*x^3*A(x)^3 + 20*x^4*A(x)^4 +...

Prog

(PARI) {a(n)=local(A=1+x, phi=(1+sqrt(5))/2); for(i=1, n, A=sum(m=0, n, x^m*(A+x*O(x^n))^floor(m*phi^2))); polcoeff(A, n)}

Crossrefs

Cf. A186576, A001950 (upper Wythoff sequence).

Sequence in context: A256332 A284715 A364473 * A141342 A232222 A241598

Adjacent sequences: A186574 A186575 A186576 * A186578 A186579 A186580

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2011

Status

approved