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A069971 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.

%I A069971 #4 Mar 30 2012 18:51:38
%S A069971 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 A069971 70,90,92,92,90,70,0,0,112,152,160,160,160,152,112,0,0,168,238,258,
%U A069971 260,260,258,238,168,0,0,240,352,392,400,400,400,392,352,240,0,0,330,498
%N A069971 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 A069971 Expected time to reach one of the boundaries at +n or -k for the first time is n*k, i.e. A004247.
%F A069971 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 A069971 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 A069971 nonn,tabl
%O A069971 0,8
%A A069971 _Henry Bottomley_, Apr 29 2002