bibkey: hanna2025a380678 authors: Paul D. Hanna year: 2025 title: “OEIS A380678, g.f. satisfying A(x - A(x)^2/(1 - A(x)^2)) = x” doi: null url: https://oeis.org/A380678 claim: “G.f. A(x) satisfies A( x - A(x)^2/(1 - A(x)^2) ) = x. Conjecture: for n>= 1, a(n) is odd iff n is a term of A027383, where A027383(2m) = 32^m - 2 and A027383(2m+1) = 42^m - 2.” strata_touched:

• D5/S1/Recurrence/Parity/QuadraticSquareReversionDyadicSupportParity license: citation-only triage: anchor

OEIS A380678

Paul D. Hanna’s entry, dated February 2025, specifies the generating equation and parity conjecture quoted above. The series has zero constant coefficient; the sequence starts at offset one and has a(n) = [x^n]A(x). The denominator has constant coefficient one, so division denotes its formal unit inverse.

The module constructs the integral solution by degree contraction and proves the stated equation, linear coefficient one, and uniqueness. In characteristic two, 1-A^2 = (1-A)^2. This identifies the reduced equation with the reversion identity of the lacunary series in D5/S1/Recurrence/Invariants/QuadraticReversionDyadicSupportParity. Uniqueness then gives its coefficient support, proving the conjecture. The formulation n+2 = 3*2^m or n+2 = 4*2^m is equivalent to the stated index families because both products are at least two.

Verified locator

• URL: https://oeis.org/A380678