Pith Number

pith:DKBHILN5

pith:2026:DKBHILN5TDV56DXYQRAQTU2HCX

not attested not anchored not stored refs resolved

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

Tong Niu

The sequence a(n) = n a(n-1) + 1 with a(0) = 0 satisfies a(n) - (n+1) a(n-1) + (n-1) a(n-2) = 0 for all n >= 2.

arxiv:2605.15500 v1 · 2026-05-15 · math.CO

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\usepackage{pith}
\pithnumber{DKBHILN5TDV56DXYQRAQTU2HCX}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

The sequence a(n) defined by a(n) = n a(n-1) + 1, a(0) = 0 satisfies a(n) - (n+1) a(n-1) + (n-1) a(n-2) = 0 for all n >= 2.

C2weakest assumption

The first-order recurrence a(k) = k a(k-1) + 1 holds exactly for every integer k >= 1, so that subtraction at adjacent indices is valid and produces no boundary or remainder terms.

C3one line summary

Three short elementary proofs establish the conjectured homogeneous recurrence for OEIS A002627 from its defining inhomogeneous recurrence.

References

11 extracted · 11 resolved · 0 Pith anchors

[1] M. Kauers and C. Koutschan,Some D-Finite and Some Possibly D-Finite Sequences in the OEIS, arXiv:2303.02793, 2023 2023

[2] M. Kauers and P. Paule,The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, Springer Texts and Monographs in Symbolic Computation, 2011. THREE SHO 2011

[3] R. J. Mathar, conjectured recurrence in OEIS sequence A002627, contributed February 16, 2014.https: //oeis.org/A002627 2014

[4] B. Salvy and P. Zimmermann,gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software20(1994), 163–177 1994

[5] Singh,OnL(m, n)and the Bernoulli and Eulerian numbers, Math 1952

Receipt and verification

First computed	2026-05-20T00:01:01.860347Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1a82742dbd98ebdf0ef8844109d34715fce10c2c8f811c4dc3a0e9ba4aa2d8cb

Aliases

arxiv: 2605.15500 · arxiv_version: 2605.15500v1 · doi: 10.48550/arxiv.2605.15500 · pith_short_12: DKBHILN5TDV5 · pith_short_16: DKBHILN5TDV56DXY · pith_short_8: DKBHILN5

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DKBHILN5TDV56DXYQRAQTU2HCX \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 1a82742dbd98ebdf0ef8844109d34715fce10c2c8f811c4dc3a0e9ba4aa2d8cb

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e2764533384f774269f767e12f34025b5d287aeb7b89a8c5e29250c2e5f24110",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-15T00:36:49Z",
    "title_canon_sha256": "e0d9c94d9e3dc39e18cd2e415b03d91da827af5d74021cf6d6d7ab95cd1d0950"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15500",
    "kind": "arxiv",
    "version": 1
  }
}