A069971 - OEIS

%I #4 Mar 30 2012 18:51:38

%S 0,0,0,0,0,0,0,2,2,0,0,8,8,8,0,0,20,22,22,20,0,0,40,48,48,48,40,0,0,

%T 70,90,92,92,90,70,0,0,112,152,160,160,160,152,112,0,0,168,238,258,

%U 260,260,258,238,168,0,0,240,352,392,400,400,400,392,352,240,0,0,330,498

%N Table by antidiagonals of variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -k for the first time.

%C Expected time to reach one of the boundaries at +n or -k for the first time is n*k, i.e. A004247.

%F T(n, k) =nk(n^2+k^2-2)/3 =T(n+1, k-1)/2+T(n-1, k+1)/2+(n-k)^2 with T(n, 0)=T(0, k)=0. T(n, n)=n^2*(n^2-1)*2/3=8*A002415(n).

%e Rows start 0,0,0,0,0,0,0,...; 0,0,2,8,20,40,70...; 0,2,8,22,48,90,152...; 0,8,22,48,92,160,258...; etc.

%K nonn,tabl

%O 0,8

%A _Henry Bottomley_, Apr 29 2002