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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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3
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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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MATHEMATICA
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CoefficientList[Series[8 (1 + x) x^2/(1 - x)^5, {x, 0, 35}], x] (* Michael De Vlieger, Jul 02 2019 *)
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PROG
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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