OFFSET
0,2
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.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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
EXAMPLE
G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 46*x^4 + 146*x^5 + 398*x^6 +...
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 *)
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 03 2014
STATUS
approved