bibkey: hanna2026a397244 authors: Paul D. Hanna year: 2026 title: “OEIS A397244, quadratic power-diagonal generating series” doi: null url: https://oeis.org/A397244 claim: “G.f. A(x) satisfies (2n) * [x^n] A(x)^(2n) = (2n-1) * [x^n] A(x)^(2n+1) for n > 1. Conjecture: for n > 0, a(n) is odd iff n = 2*A003714(k) + 1 for some k >= 0, where A003714 lists the Fibbinary numbers.” strata_touched:
- D5/S1/Recurrence/Residue/QuadraticPowerDiagonalFibbinaryParity license: citation-only triage: anchor
OEIS A397244
Paul D. Hanna’s entry specifies the NAME and COMMENT quoted verbatim in the claim above. The source snapshot attributes the entry to Paul D. Hanna, Jun 24 2026 (%A), with revision #11 dated Jun 25 2026 (%I); the citation key follows that year. The initial coefficients are a(0)=a(1)=1. The Fibbinary numbers in A003714 are the nonnegative integers with no adjacent ones in their binary expansion.
The module constructs the integer sequence by a strict-prefix recurrence:
for n > 1, put P = sum of a(j)*X^j over j < n and set
a n = (2*n-1)*coeff n (P^(2*n+1)) - (2*n)*coeff n (P^(2*n)).
The frozen diagonal multiplier proves the NAME’s equations and, by strong
induction, uniqueness over any commutative ring.
Over ZMod 2 the new even_power_diagonal lemma proves
coeff n (F^(2*n)) = 0 for every F and n > 0 by Frobenius descent.
The cubic candidate from AbsoluteReciprocalSquareFibbinaryParity therefore
satisfies A397244’s own diagonal equations. General uniqueness identifies
the reductions, and the frozen parity theorem transports to this sequence.
The integer sequences are distinct; only their reductions are identified.
The entry’s %F(2) writes the diagonal as A396846 while its own example divides by two; %F(3) writes an n+1 power while its example uses 2n+1; the composition direction in %F(4) is unverified. The formalization uses only %N and the strict-prefix recurrence. This is the erratum recorded in Library/Words/oeis2026triage0911b.md; the COMMENT is the parity claim to be proved, not an extra defining equation.
The quotes were supplied by the orchestrator in oeis-A397244.src. This seat has no network and did not read OEIS or search the literature. The source snapshot has no references and only a b-file link. The orchestrator’s finite computations were not rerun here; no literature-priority claim is made.
Verified locator
- URL: https://oeis.org/A397244