bibkey: hanna2024a378575 authors: Paul D. Hanna year: 2024 title: “OEIS A378575, g.f. satisfying A(x) = x + xA^5(x) (5-fold compositional iterate)” doi: null url: https://oeis.org/A378575 claim: “G.f. A(x) satisfies A(x) = x + xA(A(A(A(A(x))))), so that this sequence shifts left under the 5th self-COMPOSE. Conjecture: a(n) == 1 (mod 4) for n >= 1.” strata_touched:

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

OEIS A378575

Paul D. Hanna’s entry, authored December 1, 2024, defines the generating series by the five-fold compositional equation quoted above and conjectures that every positive-index coefficient is one modulo four. Here A^5 denotes composition, not multiplication. The Lean module constructs the integer series, proves its defining equation and uniqueness, and proves the conjecture as the r=5, m=4 instance of a general shifted-iterate congruence.

Verified locator

• URL: https://oeis.org/A378575