OFFSET
0,3
COMMENTS
Traditionally for the "first passage time" problems use initial condition Gf(0)=0, but here we define Gf(0)=1 to make this sequence consistent with similar sequences already present in the database.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..350 from Alois P. Heinz)
FORMULA
G.f.: 2-Pi/2*(1+4*x)/EllipticK(4*sqrt(x*(1+x))/(1+4*x)), (Maple notation).
G.f.: 2 - AGM(sqrt(1 - 8*x), 1 + 4*x). - Vaclav Kotesovec, Sep 30 2019
a(n) ~ 3*Pi*2^(3*n-1) / (n*log(n)^2) * (1 - 2*(gamma + 2*log(2) + 2*log(3)) / log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*gamma*log(3) + 24*log(2)*log(3) + 12*log(2)^2 + 12*log(3)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 30 2019
G.f.: 2 - 1/B(x) where B(x) is the g.f. of A094061. - Jesiah Darnell, Sep 22 2023
EXAMPLE
From Jesiah Darnell, Sep 22 2023: (Start)
A094061(4) - (a(1)a(3)*2 + a(2)*a(2)*1) = 216 - (0 + 64) = 152, so a(4) = 152.
A094061(7) - (a(1)a(6)*2 + a(2)*a(2)*a(3)*3 + a(2)*a(5)*2 + a(4)*a(3)*2) = 58800 - (0 + 4608 + 13056 + 7296) = 33840, so a(7) = 33840. (End)
MAPLE
G:=t->2-Pi*(1+4*t)/2/EllipticK(4*sqrt(t*(1+t))/(1+4*t)); Gf:=convert(series(G(t), t, 30), polynom): seq(print(i, coeff(Gf, t, i)), i=0..degree(Gf));
MATHEMATICA
CoefficientList[Series[2-Pi/2*(1+4*x)/EllipticK[16*x*(1+x)/(1+4*x)^2], {x, 0, 20}], x]. - Vaclav Kotesovec, Mar 10 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Sergey Perepechko, Sep 13 2004
STATUS
approved