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.

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

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

Adjacent sequences: A072816 A072817 A072818 * A072820 A072821 A072822

Keyword

nonn,easy

Author

Henry Bottomley, Jul 14 2002

Status

approved