A208887 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*(A(x) + A(-x)).

1, 1, 2, 3, 4, 6, 8, 11, 16, 22, 32, 46, 64, 92, 128, 179, 256, 358, 512, 730, 1024, 1460, 2048, 2878, 4096, 5756, 8192, 11644, 16384, 23288, 32768, 46147, 65536, 92294, 131072, 186018, 262144, 372036, 524288, 739210, 1048576, 1478420, 2097152, 2973636

Offset

0,3

Comments

Limit a(n)^(1/n) = sqrt(2).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: A(x) = (sqrt(1+4*x^4) - (1-2*x-2*x^2))/((2*x)*(1-2*x^2)).

Recurrence: (n+1)*a(n) = 2*(n+1)*a(n-2) - 4*(n-5)*a(n-4) + 8*(n-5)*a(n-6). - Vaclav Kotesovec, Aug 19 2013

a(n) ~ 2^(n/2) * (1 - 2*sin(Pi*n/4)*sin(Pi*n/2)/(sqrt(Pi)*n^(3/2))). - Vaclav Kotesovec, Aug 19 2013

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 +...
Related series:
A(x)*A(-x) = 1 + 3*x^2 + 6*x^4 + 11*x^6 + 22*x^8 + 46*x^10 + 92*x^12 +...
A(x)+A(-x) = 2 + 4*x^2 + 8*x^4 + 16*x^6 + 32*x^8 + 64*x^10 + 128*x^12 +...

Mathematica

CoefficientList[Series[(Sqrt[1+4*x^4] - (1-2*x-2*x^2))/((2*x)*(1-2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 19 2013 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*subst(A, x, -x)+x^2*(A+subst(A, x, -x+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))
(PARI) Vec((sqrt(1+4*x^4) - (1-2*x-2*x^2))/((2*x)*(1-2*x^2)) + O(x^50)) \\ G. C. Greubel, Feb 01 2017

Crossrefs

Cf. A208888, A209199, A233895, A233896.

Sequence in context: A226187 A294922 A317669 * A017911 A057048 A281094

Adjacent sequences: A208884 A208885 A208886 * A208888 A208889 A208890

Keyword

nonn

Author

Paul D. Hanna, Mar 07 2012

Status

approved