bibkey: hanna2005a107099
authors: Paul D. Hanna
year: 2005
title: "OEIS A107099, compositional half-iterate of x + 4*x^3"
doi: null
url: https://oeis.org/A107099
claim: "Coefficients [x^n] A(x) = 0 (mod 3) except at n = 3^k (conjecture)."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaHalfIterateCubicModThree
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A107099

OEIS A107099

The unpaginated Name states: “G.f. satisfies A(A(x)) = x + 4*x^3, where A(x) = Sum_{n>=0} a(n)x^(2n+1).”

The Comments state: “Coefficients [x^n] A(x) = 0 (mod 3) except at n = 3^k (conjecture).”

The Example states: “A(x) = 1x + 2x^3 - 6x^5 + 36x^7 - 266x^9 + 2028x^11 - 13596*x^13 +-…”

The series has rational coefficients, zero even coefficients and linear coefficient one. The coefficient assertion ranges over every natural exponent and refers to the integer coefficient equal to that rational coefficient. Existence and integrality are included. The statement imposes no nonzero residue at powers of three. The stronger binomial residue formula and coefficient growth are outside the formalized statement.