A208887 - OEIS

%I #20 Feb 01 2017 12:48:29

%S 1,1,2,3,4,6,8,11,16,22,32,46,64,92,128,179,256,358,512,730,1024,1460,

%T 2048,2878,4096,5756,8192,11644,16384,23288,32768,46147,65536,92294,

%U 131072,186018,262144,372036,524288,739210,1048576,1478420,2097152,2973636

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

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

%H G. C. Greubel, <a href="/A208887/b208887.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

%F 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

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 +...

%e Related series:

%e A(x)*A(-x) = 1 + 3*x^2 + 6*x^4 + 11*x^6 + 22*x^8 + 46*x^10 + 92*x^12 +...

%e A(x)+A(-x) = 2 + 4*x^2 + 8*x^4 + 16*x^6 + 32*x^8 + 64*x^10 + 128*x^12 +...

%t 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 *)

%o (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)}

%o for(n=0,60,print1(a(n),", "))

%o (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

%Y Cf. A208888, A209199, A233895, A233896.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 07 2012