Pith Number

pith:NPIGUIGP

pith:2026:NPIGUIGPHUDLEKHX73Y4WFYULS

not attested not anchored not stored refs resolved

A short proof of Mathar's 2013 recurrence conjecture for the reversible-binary-string sequence A032123

Tong Niu

Mathar's conjectured order-5 recurrence holds for the sequence counting binary strings up to reversal.

arxiv:2605.14213 v1 · 2026-05-14 · math.CO

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\usepackage{pith}
\pithnumber{NPIGUIGPHUDLEKHX73Y4WFYULS}

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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

Mathar's order-5 operator, applied to each summand separately, reduces to a polynomial identity that simplifies to zero after a brief calculation.

C2weakest assumption

The closed form a(n) = 1/2 (binomial(2n,n) + [n even] binomial(n,n/2)) correctly counts the orbits under the reversal group action, which rests on the standard application of Burnside's lemma to the two-element group.

C3one line summary

The conjectured recurrence for a(n) holds because the order-5 operator annihilates both the central binomial coefficient and the even-n middle binomial term in the closed form derived from Burnside's lemma.

References

11 extracted · 11 resolved · 1 Pith anchors

[1] Burnside,Theory of Groups of Finite Order, Cambridge University Press, 1897

[2] S. Chen, M. Kauers, C. Koutschan, X. Li, R.-H. Wang and Y. Wang,Non-minimality of minimal telescopers explained by residues, arXiv:2502.03757, 2025 2025

[3] Fried,Proofs of some conjectures from the OEIS 2024

[4] S. Fried,Proofs of several conjectures from the OEIS, J. Integer Seq.28(2025), Article 25.4.3. 10 TONG NIU 2025

[5] M. Kauers and C. Koutschan,A list of guessed but unproven holonomic recurrences in the OEIS, arXiv:2303.02793, 2023 2023

Receipt and verification

First computed	2026-05-17T23:39:10.910373Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6bd06a20cf3d06b228f7fef1cb17145ca42bff53c82ba3529c0713c6bc717db0

Aliases

arxiv: 2605.14213 · arxiv_version: 2605.14213v1 · doi: 10.48550/arxiv.2605.14213 · pith_short_12: NPIGUIGPHUDL · pith_short_16: NPIGUIGPHUDLEKHX · pith_short_8: NPIGUIGP

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/NPIGUIGPHUDLEKHX73Y4WFYULS \
  | 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: 6bd06a20cf3d06b228f7fef1cb17145ca42bff53c82ba3529c0713c6bc717db0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "4cb981462f3abdffd4857adc50ac897ea08c06eecab7281f8c033ace3513f589",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-14T00:14:56Z",
    "title_canon_sha256": "859203e54228cffbd31679dcbce8fea0291e9b17e26885afb42f63123d1bb18c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14213",
    "kind": "arxiv",
    "version": 1
  }
}