bibkey: hanna2026a396808 authors: Paul D. Hanna year: 2026 title: OEIS A396808, an implicit power-series coefficient equation doi: null url: https://oeis.org/A396808 claim: For n greater than one, residues two occur exactly at powers of three and residues one exactly at three halves of a sum of two distinct powers of three. strata_touched:

• D5/S3/Arith/TernaryTraceSupport license: citation-only triage: anchor

A396808 modulo three

The entry defines the normalized integer generating series by (n+1)[x^n]A^(n+1)=n[x^n]A^(n+2) for n>1, with a(0)=a(1)=1. Its comments state the two mod-three support claims as conjectures. The proof in TernaryTraceSupport is derived in this repository and reuses the integer sequence and its source equation from ArtinSchreierTracePowersOfTwo.

Verified locator

url: https://oeis.org/A396808

doi: null

The internal entry and its b-file were retrieved on 2026-09-10. The %N, %C and %F fields were read, and the first 18 b-file values were independently recomputed. The complete fields of the direct references A038464, A396838 and A396839 were also read. A038464 is the sequence of sums of two distinct powers of three.

Search scope and formula correction

arXiv API queries all:A396808 and all:Hanna AND (all:congruence OR all:"power series") each returned zero results. Loogle’s exact A396808 name search returned zero; GitHub code search for A396808 in Lean returned no hits. These are bounded search receipts, not a claim that no proof exists anywhere.

Let A denote A396808 and B denote A396838. A396808 formula (4)’s A=B(xA) gives second coefficient 5, conflicting with a(2)=3. The opposite direction displayed on A396838 is B=A(xB) and A=B(x/A). The formal proof does not use these inversion assertions or A396839’s self-convolution assertion. Unopened links are ASSUMED-UNVERIFIED.