bibkey: hanna2026a396099 authors: Paul D. Hanna year: 2026 title: OEIS A396099, g.f. satisfying A(x) = x + A^2(x)*A^3(x) (compositional iterates) doi: null url: https://oeis.org/A396099 claim: “G.f. satisfies A(x) = x + A^2(x)A^3(x). Conjecture: all terms are odd. Conjecture: a(n) = [1,3,3,1] repeating (mod 4) for n > 2. Conjecture: [x^n] A(A(x)) == 0 (mod 4) for n > 2. Conjecture: [x^n] ( A(x) - xA(A(A(x))) ) == 2 (mod 4) for n > 2.” strata_touched:
- D5/S1/Recurrence/Parity/IterateProductTwoThreeModFour license: citation-only triage: anchor
OEIS A396099
Paul D. Hanna’s entry, dated Jun 02 2026, states the generating equation
and the four conjectures quoted above. The superscripts in the NAME mean
compositional iterates, as specified by the entry’s FORMULA and PROG:
A^2(x) = A(A(x)) and A^3(x) = A(A(A(x))). The sequence starts at
index one with 1, 1, 5, 39, 383, 4357, 55057, 754223, 11028735.
The Lean module constructs the unique zero-constant integer solution and
proves all four conjectures. Its reduction modulo four is
F = X/(1-X) + 2*X^4/((1-X)*(1+X^2)), where division means multiplication
by the inverse of a formal series with constant coefficient one.
Clearing unit denominators proves F(F(X)) = X + 2*X^2 and
F = X + (X + 2*X^2)*(F + 2*F^2). The coefficient pattern begins at
n = 3: the residue is one when n mod 4 is two or three, and three
otherwise. Oddness is asserted for every n >= 1.
Verified locator
- URL: https://oeis.org/A396099
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