A340002 Random walk in R^3: Numerators of the expected distance after n steps.

0, 1, 4, 13, 28, 1199, 239, 113149, 1487, 14345663, 292223, 17110600987, 14849671, 545242142639, 961780559, 1704615588759647, 856088316689, 7836371329207844977, 1103759659545457, 16895087931630048788047, 59954362566895631, 2699144613568894213138579, 28918424475964028179

Offset

0,3

Comments

The random variables X_n are defined by X_0 = 0 and X_(n+1) = X_n + U_n where U_n are i.i.d. random variables with uniform distribution on the 2-dimensional sphere. Then a(n) = E(|X_n|), take numerators.

Let (V_n)_n be i.i.d. random variables with uniform distribution on the interval [-2,2]. Then a(n) = E(|V_1+...+V_n|), take numerators.

Links

Ludovic Schwob, Table of n, a(n) for n = 0..99

Formula

a(n)/A340003(n) ~ 2*sqrt(2*n)/sqrt(3*Pi).

a(n)/A340003(n) = (1/(2^(n-2) * (n+1)!)) * Sum_{k=0..floor((n-1)/2)} (-1)^k * C(n,k) * (n-2*k)^(n+1). - Ludovic Schwob, Jun 11 2022

Example

0, 1, 4/3, 13/8, 28/15, 1199/576, 239/105, 113149/46080, 1487/567, 14345663/5160960, ... = A340002/A340003.

Crossrefs

See A340003 for denominators.

Sequence in context: A155331 A155387 A155366 * A368570 A135039 A168559

Adjacent sequences: A339999 A340000 A340001 * A340003 A340004 A340005

Keyword

nonn,frac

Author

Ludovic Schwob, Dec 26 2020

Status

approved