The Seventh Compositional Iterate Modulo Eight

Abstract

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

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

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

Resolves. Problems/oeis-a396798-seventh-iterate-mod-eight (proved) by D5/S1/Recurrence/Residue/IterateProductFourFiveSeventhModEight.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 set U=I3(F). The source definition gives coeff0(F)=0 and coeff1(F)=1. The existing iterate_top theorem, comparing F with X at degree two, gives coeff2(Ij(F))=jcoeff2(F). The public third result at n=2 gives 3coeff2(F)=3; since 33=1 modulo eight, coeff2(F)=1. These low coefficients and the public fourth tail give I4(F)=X+4X^2. Iteration addition and substitution into U give I7(F)=U+4U^2. The zero constant and unit linear coefficients of U, together with the third period, give 4coeff m(U)=4 for every m>0. The convolution endpoints vanish, and its n-1 positive-index pairs each contribute four, so 4coeff n(U^2)=4(n-1) for n>1. The four cases of (n-2)%4 give 7,1,3,5; compatibility of coefficient reduction with iteration transfers this identity to integer divisibility.

References

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