Three short proofs of Mathar's 2014 conjecture for OEIS A002627

TL;DR

The paper provides three elementary proofs confirming Mathar's 2014 conjecture for OEIS sequence A002627, using homogenisation, generating functions, and binomial sum telescoping.

Contribution

It offers three novel, simple proofs of a longstanding conjecture, demonstrating elementary methods to derive the second-order recurrence.

Findings

All three proofs are elementary and require undergraduate techniques.

The homogenisation trick applies to a broader class of sequences satisfying similar recurrences.

The second-order recurrence relation holds for the sequence defined by the given recurrence.

Abstract

For the OEIS sequence A002627, defined by the inhomogeneous first-order recurrence $a(n) = n\,a(n-1) + 1$ with $a(0) = 0$, R.~J.~Mathar recorded in February 2014 the conjectured second-order homogeneous recurrence \[ a(n) - (n+1)\,a(n-1) + (n-1)\,a(n-2) = 0, \qquad n \ge 2, \] which has remained marked as a conjecture on the OEIS for over a decade. We give three short proofs. The first is two lines: subtract the defining recurrence at adjacent indices and the constant cancels (we call this homogenisation). The second reads off the same relation from the exponential generating function $F(x) = (e^x-1)/(1-x)$. The third is a Pascal-rule telescoping on the binomial-sum form $a(m) = \sum_{k=0}^{m-1} k!\binom{m}{k}$. All three derivations are elementary, requiring nothing beyond undergraduate techniques. We remark that the same homogenisation trick clears an entire class of ``Conjecture: \dots'' entries on the OEIS, namely sequences satisfying $a(n) = p(n)\,a(n-1) + q(n)$ with simple $q$.