bibkey: hanna2026a397242b authors: Paul D. Hanna year: 2026 title: “OEIS A397242, o.g.f. A = exp(x + Sum (n^2-1) a(n) x^n / n^2)” doi: null url: https://oeis.org/A397242 claim: “O.g.f. satisfies A(x) = exp( x + Sum_{n>=2} (n^2-1) * a(n)x^n / n^2 ). Conjecture: for n > 0, a(n) == 1 (mod 3) iff n+2 is of the form 3^k or 23^k. Conjecture: for n > 0, a(n) == 2 (mod 3) iff n+2 is a sum of 2 distinct powers of 3 (A038464).” strata_touched:

• D5/S1/Recurrence/Residue/ExponentialSquareWeightTernarySupport license: citation-only triage: anchor

OEIS A397242: the two ternary conjectures

Paul D. Hanna’s entry, dated June 19, 2026, gives the generating equation and the two conjectures quoted above. The sequence has offset zero. The exponents in both conjectures are nonnegative integers; distinct powers are written with exponents i<j.

The coefficient function a and its integral normalization d are those of D5/S1/Recurrence/Residue/ExponentialSquareWeightCatalanParity. That module proves a(n)=n d(n) for n>=1, the formal identity A(0)=1 and X A’=M A, the rational weights M(n)=(n^2-1)a(n)/n for n>=2, and uniqueness among rational solutions with these weights. This is the formal differential reading of the quoted exponential equation, with M=X L’ and A=exp(L).

Over ZMod(3), put U(x)=Sum d(3m+1)x^m and V(x)=Sum d(3m+2)x^m. The imported recurrence gives U=1+xVU and V=U-xV^2, hence V=1+x^2V^3. Consequently T=xV satisfies T=x+T^3 and has coefficient one exactly at powers of three. The series F=xU=T+T^2 also satisfies F=x+x^2 A modulo three. Ternary coefficient recursion classifies F: its coefficient is one at a power of three or twice a power of three, two at a sum of two distinct powers of three, and zero elsewhere. The two supports are disjoint. Shifting the index by two proves both conjectures, in both directions.

Verified locator

• URL: https://oeis.org/A397242