The Third Compositional Iterate Modulo Eight

Abstract

The third compositional iterate of A396798 has coefficient period 3,1,7,5 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 third conjecture).

$$\forall n:\mathbb{N},1<n⟹8\mid (\mathrm{coeff}\left(n,\mathrm{iterate}\left(A,3\right)\right)-(\mathrm{if}\mathrm{mod}\left((n-2),4\right)=0\mathrm{then}3\mathrm{else}\mathrm{if}\mathrm{mod}\left((n-2),4\right)=1\mathrm{then}1\mathrm{else}\mathrm{if}\mathrm{mod}\left((n-2),4\right)=2\mathrm{then}7\mathrm{else}5))$$

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

Resolves. Problems/oeis-a396798-third-iterate-mod-eight (proved) by D5/S1/Recurrence/Residue/IterateProductFourFiveThirdModEight.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 let J=I5(F). The fifth coefficient identity gives the rational form of J. Clearing unit denominators identifies its compositional inverse K=X+X^2*(3+X+7X^2+5X^3)/(1-X^4). The eighth identity I8(F)=X identifies I3(F) with K. The geometric inverse of 1-X^4 then gives the period 3,1,7,5 from degree two.

References

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