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Search: id:A072819
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| A072819 |
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Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time. |
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+0
2
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| 0, 0, 8, 48, 160, 400, 840, 1568, 2688, 4320, 6600, 9680, 13728, 18928, 25480, 33600, 43520, 55488, 69768, 86640, 106400, 129360, 155848, 186208, 220800, 260000, 304200, 353808, 409248, 470960, 539400, 615040, 698368, 789888, 890120, 999600
(list; graph; listen)
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OFFSET
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0,3
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FORMULA
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a(n) =n^2*(n^2-1)*2/3 =4*A008911(n) =8*A002415(n) =A069971(n, n).
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EXAMPLE
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a(2)=8 since for a random walk with absorbing boundaries at +2 or -2: the probability of first reaching a boundary at time t=2 is 1/2, at t=4 is 1/4, at t=6 is 1/8, at t=8 is 1/16, etc.; giving a mean of 2/2+4/4+6/8+8/16+...=4; and a variance of 2^2/2+4^2/4+6^2/8+8^2/16+...-4^2=24-16=8.
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CROSSREFS
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Cf. A000290 (i.e. n^2) for mean time. A072818(n)=sqrt(a(A001079(n))) attempts to identify the integer standard deviations.
Sequence in context: A152750 A121355 A035471 this_sequence A073912 A128796 A093199
Adjacent sequences: A072816 A072817 A072818 this_sequence A072820 A072821 A072822
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Jul 14 2002
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