OFFSET
0,3
COMMENTS
What is the limit of a(n+1)/a(n) ? Seems to be near (sqrt(5)+1)/2 ...
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..4150
Vaclav Kotesovec, Graph - a(n+1)/a(n)
FORMULA
G.f. satisfies: A(x) = A(x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + x^9) / ((1-x)*(1-x^2-x^3)).
G.f.: 1/Product_{n>=0} (1 - G(x,n)) where G(x,n+1) = G(x^2+x^3,n) with G(x,0)=x.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 14*x^6 + 23*x^7 +...
such that A(x) = A(x^2 + x^3)/(1-x), where:
A(x^2+x^3) = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 9*x^7 + 14*x^8 +
23*x^9 + 39*x^10 + 65*x^11 + 109*x^12 + 174*x^13 + 278*x^14 +...
G.f. equals an infinite product involving iterations of x^2+x^3:
A(x) = 1 / ( (1-x) * (1 - (x^2+x^3)) * (1 - (x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + x^9)) *...* (1 - G(x,n)) *...)
where G(x,n+1) = G(x^2+x^3,n) with G(x,0) = x.
Iterations of x^2+x^3 begin:
G(x,1) = x^2 + x^3;
G(x,2) = x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + x^9;
G(x,3) = x^8 + 4*x^9 + 8*x^10 + 14*x^11 + 23*x^12 + 32*x^13 + 43*x^14 + 63*x^15 + 96*x^16 + 141*x^17 + 192*x^18 + 240*x^19 + 267*x^20 + 255*x^21 + 210*x^22 + 150*x^23 + 87*x^24 + 36*x^25 + 9*x^26 + x^27;
...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, #binary(n)+1, A=subst(A, x, x^2+x^3)/(1-x +x*O(x^n))); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 23 2015
STATUS
approved