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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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FORMULA
| a(n) =n^2*(n^2-1)*2/3 =4*A008911(n) =8*A002415(n) =A069971(n, n).
G.f.: 8*(1 + x)*x^2/(1 - x)^5. [Arkadiusz Wesolowski, Feb 08 2012]
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EXAMPLE
| 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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PROG
| (MAGMA) [n^2*(n^2-1)*2/3: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011
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CROSSREFS
| 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: A121355 A168012 A035471 * A190317 A073912 A187174
Adjacent sequences: A072816 A072817 A072818 * A072820 A072821 A072822
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jul 14 2002
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