OFFSET
0,3
COMMENTS
All terms appear to be odd.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..730
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)^2 + x^2*(1-x)*A(x)^3.
(2) 1 - x = Sum_{n>=0} (-1)^(n mod 3) * x^n * A(x)^n.
(3) A(x) = (1 - x^3*A(x)^3)/(1 - x*A(x) - x^2*A(x)^2).
(4) A(x) = Series_Reversion( x*(1 - x - x^2)/(1 - x^3) ).
(5) A( x*(1 - x - x^2)/(1 - x^3) ) = (1 - x^3)/(1 - x - x^2).
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 127*x^5 + 517*x^6 + 2173*x^7 + 9379*x^8 + 41293*x^9 + 184765*x^10 + ...
where A(x) = 1 + x*A(x)^2 + x^2*(1-x)*A(x)^3.
Also,
1 - x = 1 - x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 - x^4*A(x)^4 + x^5*A(x)^5 + x^6*A(x)^6 - x^7*A(x)^7 + x^8*A(x)^8 +-+ ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 7*x^2 + 24*x^3 + 93*x^4 + 374*x^5 + 1567*x^6 + 6736*x^7 + 29581*x^8 + 132070*x^9 + 597755*x^10 + ...
A(x)^3 = 1 + 3*x + 12*x^2 + 46*x^3 + 189*x^4 + 795*x^5 + 3438*x^6 + 15150*x^7 + 67845*x^8 + 307785*x^9 + 1411638*x^10 + ...
Let F(x) satisfy A(x/F(x)) = F(x) then
F(x) = 1 + x + 2*x^2*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ... + Fibonacci(n+1)*x^n + ...).
PROG
(PARI) {a(n) = my(A=1); A = (1/x)*serreverse( x*(1 - x - x^2)/(1 - x^3 +x*O(x^n)) ); polcoef(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 20 2025
STATUS
approved