OFFSET
0,3
COMMENTS
Conjecture: a(n) is odd when n = (4^k - 1)/3 for k >= 0, and even elsewhere.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..600
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^4 + A(-x)^4)/2.
(2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2.
(3) A(x) = A(-x) + x*A(x)^4 + x*A(-x)^4.
(4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).
(5) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 32*x^4 + 345*x^5 + 810*x^6 + 10492*x^7 + 24880*x^8 + 356252*x^9 + 848992*x^10 + 12946094*x^11 + 30942208*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 32*x^3 + 96*x^4 + 810*x^5 + 2634*x^6 + 24880*x^7 + 84668*x^8 + 848992*x^9 + 2974649*x^10 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 84*x^3 + 345*x^4 + 2324*x^5 + 10492*x^6 + 74540*x^7 + 356252*x^8 + 2609552*x^9 + 12946094*x^10 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 2*x^2 + 32*x^4 + 810*x^6 + 24880*x^8 + 848992*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 32*x^3 + 810*x^5 + 24880*x^7 + 848992*x^9 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^4:
(A(x) - A(-x))/2 = x + 14*x^3 + 345*x^5 + 10492*x^7 + 356252*x^9 + ...
(A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 345*x^4 + 10492*x^6 + 356252*x^8 + ...
PROG
(PARI) {a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 - B^2)/2 + x*(A^4 + B^4)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 10 2024
STATUS
approved