Build the generating series
Package the lattice-walk counts into a formal power series so the whole sequence can be studied at once.
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INTERACTIVE RESEARCH GUIDE
Three coordinates. One exact sequence.
Separate the constraints of a three-dimensional walk, then count their interleavings.
Candidate resolutionA Bessel-Factorization Proof of Mathar's Recurrence for Type-ace Lattice Walks
Three different rules
Coordinate a stays nonnegative and returns to zero. Coordinate c stays nonnegative but may end anywhere. Coordinate e is unrestricted.
- Length
- 6
- Type-ace walks
- 3911
- Recurrence residual
- 0
The exact coordinate formula gives a6 = 3,911. Its recurrence residual is 0.
aₙ = Σ n!/(r!s!t!) · Aᵣ · Cₛ · 2ᵗ, with r+s+t=n
TRY THIS
Set length to 4: the coordinate formula gives 188, and the recurrence residual is exactly zero.
What this experiment represents. The coordinate-count formula and recurrence follow MATH-2026-09. Counts in the displayed range are exact integers; the Bessel factorization and general proof are not re-proved here.
Mathematics · Preprint
A Bessel-Factorization Proof of Mathar's Recurrence for Type-ace Lattice Walks
A recurrence proof with exact Ore-operator verification
Candidate resolutionMF-PRISM-MATH-2026-09 / v1.1
A proposed proof of Mathar's conjectured recurrence for the Type-ace lattice-walk sequence OEIS A302186. Exact verification covers coefficients through n=100 and the Ore-operator identity.
- Identifier
- MF-PRISM-MATH-2026-09
- Series
- Mathematics
- Edition
- Version 1.1
- Length
- 9 pages
01
Plain-language guide
3 minute explanation
A plain-language summary.
The paper explains a conjectured lattice-walk recurrence by factoring the generating function into Bessel-function pieces and transferring that structure into an exact recurrence operator.
lattice walksbessel functionsrecurrencesexact computation
A recurrence predicts each term of a sequence from earlier terms. Mathar observed such a rule for a Type-ace lattice-walk sequence, but a numerical pattern is not yet an explanation.
This paper’s proposed explanation passes through special functions. It rewrites the walk-generating series using Bessel factors, derives the differential or operator relation they satisfy, and converts that relation back into the conjectured recurrence.
02
How it works
The argument, without the notation.
Factor through Bessel functions
Express the series using Bessel-type components whose differential behavior is already structured.
Transfer to a recurrence
Use an Ore-operator identity to convert the factorization into the claimed coefficient recurrence, then verify coefficients exactly through n = 100.
03
What it adds
Main findings.
The conjectured recurrence is given a structural derivation rather than only a numerical fit.
The factorization, formal-series coefficients, and recurrence operator are checked exactly.
The manuscript targets OEIS A302186 and does not claim a recurrence theorem for unrelated lattice-walk models.
04
Scope & limits
Evidence and limitations.
Current status
Proposed proof released for independent specialist and publication-priority review.
This manuscript presents a candidate resolution. Publication is not verification: independent specialist review remains necessary before the result should be treated as established.
Exact verification to a large finite order supports the proof but cannot replace review of the symbolic derivation.
The claim is specific to the Type-ace sequence and its stated generating function.
Novelty and publication priority remain separate from mathematical correctness.
05
Why it matters
Why this research matters.
A proof would replace a conjectured numerical pattern with a structural explanation tied to special functions and operator factorization.
06
Key terms
Terms used in this paper.
These are the terms needed to understand the claim. The full paper uses them more precisely.
Lattice walk
A walk whose steps follow allowed moves on a regular grid.
Generating function
A formal series that stores an entire sequence in its coefficients.
Bessel function
A classical special function satisfying a structured differential equation.
Ore operator
An algebraic operator used to represent differential or recurrence relations exactly.
07
Independent review
Where scrutiny should concentrate.
- 01The Bessel-factorization derivation.
- 02The Ore-operator identity and recurrence transfer.
- 03The distinction between exact finite verification and the general proof, including priority review.
Publication record
Read the paper and examine the source files.
This page is a reading guide, not a substitute for the manuscript. The public record links the explanation to the paper, source package, review materials, and persistent identifier.
- Document
- Preprint
- Review status
- Open for independent review
- Published
- 2026-07-18
- DOI status
- Published · 10.5281/zenodo.21434724
- Canonical record
- MF-PRISM-MATH-2026-09
Metriq PRISM Laboratory, A Bessel-Factorization Proof of Mathar's Recurrence for Type-ace Lattice Walks, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-09, Version 1.1, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.21434724.