A186577 - OEIS

%I #10 Mar 30 2012 18:37:26

%S 1,1,3,13,65,353,2025,12074,74083,464708,2966647,19211268,125890754,

%T 833242554,5562338802,37406660537,253185542824,1723428340232,

%U 11790556480270,81026845851661,559086953351167,3871831119088196

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

%F 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].

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 65*x^4 + 353*x^5 + 2025*x^6 +...

%e The g.f. satisfies:

%e 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) +...

%e The g.f. of A186576, F(x) = A(x/F(x)), satisfies:

%e 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) +...

%e and begins:

%e F(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 72*x^5 + 274*x^6 +...

%e Since A(x) = F(x*A(x)), then:

%e A(x) = 1 + x*A(x) + 2*x^2*A(x)^2 + 6*x^3*A(x)^3 + 20*x^4*A(x)^4 +...

%o (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)}

%Y Cf. A186576, A001950 (upper Wythoff sequence).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 24 2011