A248169 G.f.: AGM( (1-x)/(1+x), (1+3*x)/(1-3*x) ).

1, 2, 6, 18, 46, 146, 398, 1234, 3454, 10610, 30230, 92290, 265670, 808210, 2343182, 7110994, 20720798, 62774578, 183617910, 555559554, 1629845726, 4926370114, 14486707710, 43752525762, 128909169334, 389075915954, 1148184115062, 3463595977986, 10235096852902

Offset: 0

Author

Paul D. Hanna, Oct 03 2014

Keywords

• nonn

Comments

Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

a(n) == 2 (mod 4) for n>0.

Limit a(n+1)/a(n) = 3.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 46*x^4 + 146*x^5 + 398*x^6 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A063887.

Programs

Mathematica

CoefficientList[Series[Pi*(1 + 3*x)/((2*(1 - 3*x)*EllipticK[(16*x*(1 + 3*x^2)) / ((1 + x)^2*(1 + 3*x)^2)])), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 26 2019 *)

PARI

{a(n)=local(A,X=x+x*O(x^n));A=agm((1-x)/(1+X),(1+3*x)/(1-3*X)); polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

Formula

a(n) ~ Pi * 3^n / log(n) * (1 - (gamma + 4*log(2))/log(n) + (gamma^2 + 8*log(2)*gamma + 16*log(2)^2 - Pi^2/6) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 30 2019