bibkey: hanna2026a392203 authors: Paul D. Hanna year: 2026 title: “OEIS A392203, G.f. A(x) satisfies A(x - A(x)) = x^2 + xA(x)” doi: null url: https://oeis.org/A392203 claim: “A392203 %C: It appears that a(2n) == 1 (mod 8) and a(2*n+1) == 3 (mod 8) for n >= 1.” strata_touched:

• D5/S1/Recurrence/Invariants/HannaShiftSquareModEight license: citation-only triage: anchor

OEIS A392203

Paul D. Hanna’s entry of January 3, 2026 defines A(x) = Σ_{n≥2} a(n) x^n by A(x - A(x)) = x^2 + x*A(x). Its terms begin 1, 3, 17, 131, 1209, 12603, 143705, 1757491, 22757185, 309200275, 4379508569, ….

Verified locator

• URL: https://oeis.org/A392203
• Locator: NAME, “G.f. A(x) satisfies A(x - A(x)) = x^2 + xA(x).”; COMMENTS, “It appears that a(2n) == 1 (mod 8) and a(2*n+1) == 3 (mod 8) for n >= 1.”
• Sequence author Paul D. Hanna, Jan 03 2026; revision 12, Jan 05 2026. The comment carries no answer.

Scope of the recorded answer

The comment holds, and for a structural reason: modulo 8 the generating function is the rational function (x^2 + 3x^3)/(1 - x^2), whose coefficients are 1 at even and 3 at odd indices from two on. Put R = (x^2 + 3x^3)/(1 - x^2), D = 1 - x^2 and N = x - x^2 - 4x^3, so that x - R = N/D. Then R(x - R) = x^2 + x*R modulo 8 reduces to the polynomial congruence N^2*D + 3*N^3 ≡ (x^2*D + x^3 + 3x^4)*(D^2 - N^2) (mod 8), whose two sides differ by a polynomial with every coefficient divisible by 8; D and D^2 - N^2 have constant term one and are units. The equation determines its solution uniquely over any commutative ring: the map T(f) = x^2 + x*f - (f(x - f) - f) sends two series of order at least two that agree below degree d to series agreeing below degree d + 1. Hence the reduction of the integer solution is R. The same contraction, iterated from zero over the integers, produces the solution.

Bounded prior-resolution evidence

The entry was read in full and records no proof and no reference to one. Its only cross-reference, A276370 (A(x - A(x)) = x^2), states no congruence for this sequence. Searching the catalogue by the first seven terms returns only this entry, and google-deepmind/formal-conjectures has no file for it. Citation indices were not exhaustively reachable, so this is a bounded negative finding and no worldwide priority claim is made.