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.

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, 70, 90, 92, 92, 90, 70, 0, 0, 112, 152, 160, 160, 160, 152, 112, 0, 0, 168, 238, 258, 260, 260, 258, 238, 168, 0, 0, 240, 352, 392, 400, 400, 400, 392, 352, 240, 0, 0, 330, 498

Offset

0,8

Comments

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

Links

Table of n, a(n) for n=0..68.

Formula

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).

Example

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.

Crossrefs

Sequence in context: A175917 A320525 A357647 * A167291 A063865 A230275

Adjacent sequences: A069968 A069969 A069970 * A069972 A069973 A069974

Keyword

nonn,tabl

Author

Henry Bottomley, Apr 29 2002

Status

approved