A098070 Consider a single king on an infinite chessboard. This sequence gives number of n-move paths when king starting at origin reaches the origin again for the first time at step n.

1, 0, 8, 24, 152, 816, 5320, 33840, 229144, 1560864, 10906576, 76962912, 550406472, 3969725856, 28875757200, 211436151456, 1557623566104, 11533972310976, 85802992349344, 640901090847360, 4804716170926672, 36138383022850368, 272621594933332000

Offset: 0

Author

Sergey Perepechko, Sep 13 2004

Keywords

• nonn

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.

Examples

			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)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..350 from Alois P. Heinz)

Crossrefs

Cf. A094061, A054474.

Programs

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,0,22}],x] (* Vaclav Kotesovec, Mar 10 2014 *)

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