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

Abstract

For the OEIS sequence A002627, defined by the inhomogeneous first-order recurrence $a(n)=na(n-1)+1$ 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 $a(m)={\sum }_{k=0}^{m-1}k!\left(\genfrac{}{}{0ex}{}{m}{k}\right)$. 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 $q$.

arXiv comment11 pages