bibkey: mathar2012a026023 authors: R. J. Mathar year: 2012 title: “OEIS A026023, walks on the nonnegative integers from 3: recurrence conjecture” doi: null url: https://oeis.org/A026023 claim: “The entry defines a(n) as the number of sequences s(0), …, s(n) of nonnegative integers with |s(i) - s(i-1)| = 1 and s(0) = 3, records that a(n)/2^n is the probability that a random walker started at x = 4 is not adsorbed at x = 0 by time n, and records Mathar’s conjecture (n+4)(n-1)a(n) + (n-1)(n+1)a(n-1) - 2(n+1)(2n+1)a(n-2) - 4(n-1)(n+1)*a(n-3) = 0.” strata_touched:

• D5/S3/StatisticalMechanics/RandomWalks/SurvivingWalkRecurrence license: citation-only triage: anchor

OEIS A026023

A026023 (Clark Kimberling, offset 0, data 1, 2, 4, 8, 15, 30, 56, 112, …) is

a(n) = number of (s(0), s(1), …, s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,…,n and s(0) = 3.

A comment by Robert M. Ziff (2014) reads

a(n)/2^n is the probability that a random walker starting at x=4 and jumping +-1 with equal probability at each time step is not adsorbed at the boundary x=0 at time n.

and the formula field contains a(2n) = C(2n+2, n), a(2n+1) = 2*a(2n). and

Conjecture: (n+4)(n-1)a(n) +(n-1)(n+1)a(n-1) -2(n+1)(2n+1)a(n-2) -4(n-1)(n+1)*a(n-3)=0. - R. J. Mathar, Sep 29 2012

Verified locator

• URL: https://oeis.org/A026023 (revision 47, last modified 2025-10-13, name, comment and formula fields; retrieved 2026-09-27 through the OEIS JSON interface).