The Fifth Compositional Iterate Modulo Eight

Abstract

The fifth compositional iterate of A396798 has coefficient period 5,1,1,5 modulo eight from degree two.

Let A denote D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries, the unique integer ordinary formal power series with zero constant coefficient satisfying A=X+I4(A)*I5(A). Write I0(F)=X and I(k+1)(F)=Ik(F) composed with F. The product is ordinary series multiplication, with no factorial scaling.

Theorem 1.1 (Hanna’s fifth conjecture).

$$\forall n:\mathbb{N},1<n⟹8\mid (\mathrm{coeff}\left(n,\mathrm{iterate}\left(A,5\right)\right)-(\mathrm{if}(\mathrm{mod}\left((n-2),4\right)=0\vee \mathrm{mod}\left((n-2),4\right)=3)\mathrm{then}5\mathrm{else}1))$$

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

Resolves. Problems/oeis-a396798-fifth-iterate-mod-eight (proved) by D5/S1/Recurrence/Residue/IterateProductFourFiveFifthModEight.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Commentary.

Reduce A modulo eight to F, and write P=I4(F) and J=I5(F). The fourth and eighth iterate identities give P=X+4X^2 and I8(F)=X. Substituting P into F=X+PJ gives J=P+XF. Elimination yields (J-X)(1-X^4)=5X^2+X^3+X^4+5X^5. The geometric inverse of 1-X^4 has coefficient one at multiples of four and zero elsewhere, so the stated period starts at degree two.

References

• Truth anchor: D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries
• Truth anchor: D5/S1/Recurrence/Residue/IterateProductFourFiveFifthModEight.result
• Dependency: D5/S1/Recurrence/Residue/IterateProductFourFiveFourthModEight