OFFSET
0,3
COMMENTS
Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1200
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) = A(-x) + x*A(x)^2 + x*A(-x)^2.
(3) A(x) = 2 - A(-x) + x*A(x)^4 - x*A(-x)^4.
(4) A(x) = 2 - A(-x) + (A(x) - A(-x))*(A(x)^2 - A(-x)^2).
(5.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(5.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(6) (A(x) + A(-x))/2 = 1/(1 - (A(x) - A(-x))^2).
(7.a) Sum_{n>=0} a(n) * (sqrt(2) - 1)^n/3^n = sqrt(2).
(7.b) Sum_{n>=0} a(n) * (1 - sqrt(2))^n/3^n = 1.
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 88*x^4 + 210*x^5 + 2644*x^6 + 6493*x^7 + 91992*x^8 + 229646*x^9 + 3484008*x^10 + 8789562*x^11 + 139443168*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 210*x^4 + 668*x^5 + 6493*x^6 + 21538*x^7 + 229646*x^8 + 779772*x^9 + 8789562*x^10 + ...
A(x)^4 = 1 + 4*x + 22*x^2 + 88*x^3 + 605*x^4 + 2644*x^5 + 20114*x^6 + 91992*x^7 + 741154*x^8 + 3484008*x^9 + 29125100*x^10 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 9*x^3 + 210*x^5 + 6493*x^7 + 229646*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 210*x^4 + 6493*x^6 + 229646*x^8 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^4:
(A(x) + A(-x))/2 = 1 + 4*x^2 + 88*x^4 + 2644*x^6 + 91992*x^8 + 3484008*x^10 + ...
(A(x)^4 - A(-x)^4)/2 = 4*x + 88*x^3 + 2644*x^5 + 91992*x^7 + 3484008*x^9 + ...
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071....
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