The Fifth-Iterate Exponential Pattern Modulo Three

Abstract

From index three, the coefficients of OEIS A396805 have repeating residues 0, 1, 0 modulo three.

Let A be the unique rational formal series with zero constant coefficient satisfying A=X*exp(A^{[5]}), where A^{[5]} is the fifth compositional iterate. Write a(n)=n![X^n]A. The existing A_equation and fixed_unique in IterateExponentialParity identify A with AK(5) and a(n) with aK(5,n). In the formula, ite(p,x,y) equals x when p holds and y otherwise.

Theorem 1.1 (Hanna’s pattern from index three).

$$\forall n:\mathbb{N},3\le n⟹\mathrm{aK}\left(5,n\right)\mathrm{mod}3=\mathrm{ite}\left(n\mathrm{mod}3=1,1,0\right)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/IterateExponentialFiveModThree.result (✓ std3). ∎

Resolves. Problems/oeis-a396805-iterate-exponential-mod-three (proved) by D5/S1/Recurrence/Residue/IterateExponentialFiveModThree.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A396805: iterated-exponential congruences at iterate count 5. URL: https://oeis.org/A396805.

Commentary.

The residue is one precisely at indices congruent to one modulo three and zero at the other indices in the domain n>=3. This is the source pattern 0, 1, 0 starting at n=3. The source value a(2)=2 explains why that lower bound must be retained. The earlier parity result and the separately recorded non-kernel modulo-five counterexamples have their own evidence; this theorem resolves only the modulo-three clause.

References

• Truth anchor: D5/S1/Recurrence/Residue/IterateExponentialFiveModThree.result
• Dependency: D5/S1/Recurrence/Residue/IterateExponentialParity