bibkey: mathar2012a068551 authors: R. J. Mathar year: 2012 title: “OEIS A068551, a(n) = 4^n - binomial(2n, n): recurrence conjecture” doi: null url: https://oeis.org/A068551 claim: “The entry defines a(n) = 4^n - binomial(2n, n), records that it counts the returns to the x axis in all lattice paths with steps (1,1) and (1,-1) from the origin to (2n, 0), and records Mathar’s conjecture na(n) + 2(3-4n)a(n-1) + 8(2n-3)*a(n-2) = 0.” strata_touched:

• D5/S3/Combinatorics/LatticeReturnsRecurrence license: citation-only triage: anchor

OEIS A068551

A068551 is “a(n) = 4^n - binomial(2*n,n).” (offset 0, data 0, 2, 10, 44, 186, 772, …). A comment by Geoffrey Critzer (2012) reads

Total number of returns to the x axis in all lattice paths using steps (1,1) and (1,-1) from the origin to (2n,0).

and the formula field contains

Conjecture: na(n) + 2(3-4n)a(n-1) + 8(2n-3)*a(n-2) = 0. - R. J. Mathar, Apr 01 2012

Verified locator

• URL: https://oeis.org/A068551 (revision 88, last modified 2026-05-30, formula field; retrieved 2026-09-27 through the OEIS JSON interface).