Metadata
bibkey: bala2021triangularcuberoots
authors: Peter Bala; Paul D. Hanna
year: 2021
title: OEIS A193237 and A193236, congruences for triangular reciprocal cube roots
doi: null
url: https://oeis.org/A193237/internal
claim: "Conjectures: a(5*n+4) == 0 (mod 5) and a(7*n+5) == 0 (mod 7) (both checked up to n = 200). - _Peter Bala_, Feb 26 2021"
strata_touched:
- D5/S1/Recurrence/Residue/BalaA193237ModFiveSeven
- D5/S1/Recurrence/Residue/BalaA193236ModFiveSeven
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A193237/internal
Companion source: https://oeis.org/A193236/internal
The source has no DOI, arXiv identifier or page number. A193237 revision 18 and A193236 revision 20 share the revision timestamp Sep 22 2025 16:01:08. Their unnumbered FORMULA lines attribute both conjectures to Peter Bala; Paul D. Hanna authored both sequence entries.
Source statements
A193237’s NAME line reads verbatim:
Expansion of g.f.: ( Sum_{n>=0} (2n+1)3^n(-x)^(n(n+1)/2) )^(-1/3).
A193236’s NAME line reads verbatim:
Expansion of g.f.: ( Sum_{n>=0} (-3)^n*(2*n+1)x^(n(n+1)/2) )^(-1/3).
Each entry’s FORMULA line reads verbatim:
Conjectures: a(5n+4) == 0 (mod 5) and a(7n+5) == 0 (mod 7) (both checked up to n = 200). - Peter Bala, Feb 26 2021
Both offsets are zero. The examples choose the reciprocal cube root with constant coefficient one. The Lean definitions use rational formal power series and the literal weights; each coefficient is a finite sum over j in Finset.range(m+1). The claim asserts existence, integral coefficients for every normalized rational solution, and both divisibilities for every natural n, without a finite index bound.
Mathematical scope
For F=1+9H over the integers with constantCoeff(H)=0, coefficient contraction constructs U=H-3U²-3U³. Thus B=1+3U satisfies B³=F, and its inverse is integral. For any integer weights w, the series with coefficient (2j+1)w(j) at triangular degree j(j+1)/2 is supported in residues {0,1} modulo five and {0,1,3} modulo seven. Its cube misses residue four modulo five; its square misses residue five modulo seven. Frobenius transfers these gaps to an integral reciprocal cube root. Rational uniqueness covers every normalized solution.
Finite evidence and literature boundary
The independent exact-arithmetic experiment is in
the-omega-institute/trureturing-experiments, commit
67ccba7d1caa3774156d523b819df3316774e695, directory
docs/reports/bala-a193237-a193236-mod5-mod7/. The SHA-256 of check.py is
7613bcfbf58f29e849b82b9e80a311bf05c535445c74477217e58e8e83a44ad7.
The reported command python3 /tmp/op-bala193237/check.py --bfile-dir /tmp/op-bala193237/src
exited 0 with final line ALL_CHECKS_OK. It checked both b-files through
index 1000, 701 mod-five cases and 501 mod-seven cases per source through
degree 3505. These are reported finite checks, independent of the Lean proof.
Literature completeness is ASSUMED-UNVERIFIED. Issues https://github.com/the-omega-institute/trureturing/issues/22112 and https://github.com/the-omega-institute/trureturing/issues/22113 record the screened scope and rate-limited searches. No absence-of-prior-proof claim follows from those limits. Other moduli, higher prime powers and asymptotics remain open here; no downstream source theorem relying on these conjectures was identified in the recorded screen.