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A072819 Variance of time for a random walk starting at 0 to reach one of the boundaries at +n or -n for the first time. 3
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; text; internal format)
OFFSET
0,3
LINKS
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
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
E.g.f.: 2*exp(x)*x^2*(6 + 6*x + x^2)/3. - Stefano Spezia, Dec 12 2021
a(n) = 2*n * A007290(n+1). - C.S. Elder, Jan 09 2024
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.
MATHEMATICA
CoefficientList[Series[8 (1 + x) x^2/(1 - x)^5, {x, 0, 35}], x] (* Michael De Vlieger, Jul 02 2019 *)
PROG
(Magma) [n^2*(n^2-1)*2/3: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011
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: A280056 A035471 A209443 * A190317 A073912 A212571
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jul 14 2002
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)