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A069971
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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.
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1
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Expected time to reach one of the boundaries at +n or -k for the first time is n*k, i.e. A004247.
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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).
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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.
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CROSSREFS
| Sequence in context: A119332 A089262 A175917 * A167291 A063865 A037224
Adjacent sequences: A069968 A069969 A069970 * A069972 A069973 A069974
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KEYWORD
| nonn,tabl
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Apr 29 2002
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