Three short proofs of Mathar's 2014 conjecture for OEIS A002627
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arXiv
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2026
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| author | Niu, Tong |
| author_facet | Niu, Tong |
| contents | 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 $q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_15500 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Three short proofs of Mathar's 2014 conjecture for OEIS A002627 Niu, Tong Combinatorics 11B37, 11B83, 05A19 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 $q$. |
| title | Three short proofs of Mathar's 2014 conjecture for OEIS A002627 |
| topic | Combinatorics 11B37, 11B83, 05A19 |
| url | https://arxiv.org/abs/2605.15500 |