A368634 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 - A(-x)^2)/2.

1, 1, 4, 9, 52, 138, 904, 2581, 18020, 53622, 389112, 1189146, 8855560, 27571156, 209174544, 660249549, 5079702852, 16203796158, 126033559960, 405408758062, 3180991167640, 10301855821452, 81414086371696, 265150389430914, 2108026107021224, 6897985805906972, 55119920086104496

Offset: 0

Author

Paul D. Hanna, Jan 12 2024

Keywords

• nonn

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0 and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 52*x^4 + 138*x^5 + 904*x^6 + 2581*x^7 + 18020*x^8 + 53622*x^9 + 389112*x^10 + 1189146*x^11 + 8855560*x^12 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 138*x^4 + 452*x^5 + 2581*x^6 + 9010*x^7 + 53622*x^8 + 194556*x^9 + 1189146*x^10 + 4427780*x^11 + 27571156*x^12 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 9*x^3 + 138*x^5 + 2581*x^7 + 53622*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 138*x^4 + 2581*x^6 + 53622*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 4*x^2 + 52*x^4 + 904*x^6 + 18020*x^8 + 389112*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 26*x^3 + 452*x^5 + 9010*x^7 + 194556*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + 2*x * (A(x)^2 - A(-x)^2)/2.
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(3) at r = (sqrt(3) - 1)/4 = 0.183012701892219....
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..600

Crossrefs

Cf. A368633, A368629, A368627.

Programs

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 + 2*x*(A^2 - B^2)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = my(A); A = (1/x)*serreverse( (1 + 6*x - 8*x^2 - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)))/4 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1.a) A(x) = 1 + x*(3*A(x)^2 - A(-x)^2)/2.

(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 2*x*(A(x)^2 - A(-x)^2)/2.

(2.a) (A(x) + A(-x))/2 = 1 + 2*x*(A(x)^2 - A(-x)^2)/2.

(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.

(2.c) (A(x) + A(-x))/2 = 1/(1 - 2*x*(A(x) - A(-x))).

(3.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).

(3.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).

(4.a) A(x) = (1 - sqrt(1-16*x + 8*x*A(-x) + 16*x^2*A(-x)^2)) / (4*x).

(4.b) A(-x) = (sqrt(1+16*x - 8*x*A(x) + 16*x^2*A(x)^2) - 1) / (4*x).

(5) 0 = (1-2*x) - (1-6*x)*A(x) - x*(3+8*x)*A(x)^2 + 12*x^2*A(x)^3 - 8*x^3*A(x)^4.

(6) x = (1 + 6*x*A(x) - 8*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/4.

(7) A(x) = (1/x)*Series_Reversion( (1 + 6*x - 8*x^2 - sqrt(1 + 4*x - 4*x^2))/4 ).

(8.a) Sum_{n>=0} a(n) * (sqrt(3) - 1)^n/4^n = sqrt(3).

(8.b) Sum_{n>=0} a(n) * (1 - sqrt(3))^n/4^n = 1.