bibkey: hanna2026a396798
authors: Paul D. Hanna
year: 2026
title: "OEIS A396798: compositional iterates modulo eight"
doi: null
url: https://oeis.org/A396798
claim: "The fourth and eighth iterates have vanishing tails modulo eight; the third and fifth have periods 3,1,7,5 and 5,1,1,5 from degree two. The seventh has period 7,1,3,5 from degree two."
strata_touched:
  - D5/S1/Recurrence/Residue/IterateProductFourFiveSeventhModEight
  - D5/S1/Recurrence/Residue/IterateProductFourFiveThirdModEight
  - D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight
  - D5/S1/Recurrence/Residue/IterateProductFourFiveFourthModEight
  - D5/S1/Recurrence/Residue/IterateProductFourFiveFifthModEight
license: citation-only
triage: anchor

OEIS A396798: compositional iterates modulo eight

Paul D. Hanna’s June 16, 2026 entry defines the ordinary integer series A(X)=Sum_{n>=1} a(n)X^n by A=X+I4(A)*I5(A), where Ik is the k-fold compositional iterate and the product is ordinary multiplication. There is no factorial scaling. Revision 12, dated June 17, states these comments:

Third:

Conjecture: [x^n] A^3(x) == [3,1,7,5] repeating (mod 8) for n > 1.

Fourth:

Conjecture: [x^n] A^4(x) == 0 (mod 8) for n > 2.

Fifth:

Conjecture: [x^n] A^5(x) == [5,1,1,5] repeating (mod 8) for n > 1.

Seventh:

Conjecture: [x^n] A^7(x) == [7,1,3,5] repeating (mod 8) for n > 1.

Eighth:

Conjecture: [x^n] A^8(x) == 0 (mod 8) for n > 1.

IterateProductFourFiveEighthModEight.generatingSeries is the common zero-constant integer source, constructed by coefficient stabilization. IterateProductFourFiveEighthModEight.result proves the eighth statement. IterateProductFourFiveFourthModEight.result and IterateProductFourFiveFifthModEight.result use that same definition and prove the fourth and fifth statements, respectively. The fifth period starts at n=2: residues 0 or 3 of n-2 select 5, and residues 1 or 2 select 1. IterateProductFourFiveThirdModEight.result uses the same source and proves the third period: residues 0,1,2,3 of n-2 select 3,1,7,5. IterateProductFourFiveSeventhModEight.result separately settles the seventh statement on that same source for every n>1: residues 0,1,2,3 of (n-2)%4 select 7,1,3,5, beginning at n=2.

In the fourth and eighth arguments, the source equation and uniqueness identify its reduction modulo four with X/(1-X). For the fourth iterate, the second iterate reduces to X+2X^2 modulo four. An exact integer quotient writes its reduction modulo eight as G=X+2X^2+4B. Substitution annihilation and ring identities give G composed with G=X+4X^2. The eighth argument instead writes the fourth iterate as X+4B and uses square-zero iteration.

For the fifth iterate, let F be A modulo eight, P=I4(F) and J=I5(F). The two existing results and source-derived low coefficients give P=X+4X^2 and I8(F)=X. Substituting P into F=X+PJ gives J=P+XF. Eliminating F and multiplying out in characteristic eight yields (J-X)(1-X^4)=5X^2+X^3+X^4+5X^5. The geometric inverse of the unit 1-X^4 then gives the fifth period for every n>1. No proposed full formula for A or unproved second-iterate strong congruence is a premise.

For the third iterate, reconstruct J=I5(F) from the public fifth coefficients and its zero constant and unit linear coefficients. Over ZMod8, the series K=X+X^2*(3+X+7*X^2+5*X^3)/(1-X^4) satisfies K composed with J=X: an exact polynomial identity follows after clearing the unit denominators. Since I8(F)=X, compositional cancellation identifies I3(F)=K, and coefficient extraction gives the third period.

For the seventh iterate, set U=I3(F). The source definition gives coeff0(F)=0 and coeff1(F)=1. The existing iterate_top theorem compares F with X at degree two to give coeff2(Ij(F))=jcoeff2(F). The public third result at n=2 gives 3coeff2(F)=3; since 33=1 in ZMod8, coeff2(F)=1. These low coefficients and the public fourth tail give I4(F)=X+4X^2. Iteration addition and the zero constant coefficient of U give I7(F)=U+4U^2. The third period and coeff1(U)=1 imply 4coeff_m(U)=4 for m>0. The convolution endpoints vanish and its n-1 positive pairs each contribute four, giving 4coeff_n(U^2)=4(n-1). The four cases of (n-2)%4 yield the seventh period. All new adaptations are local to result; the existing generic declarations used include accessible private declarations.

The third, fourth, fifth and eighth statements were preregistered in #7982, #7950, #7961 and #7943, respectively. The original first comment’s n>=1 period conflicts with its own a(3)=9; that previously recorded non-kernel observation is not a new formal refutation. The seventh statement is preregistered separately in #7994 before proof or numerical probes. Comments 2 and 6 remain with #6436 (and the #6375 referral); a proposed correction of the first comment retains its separate obligation. The bounded source, cross-reference and exact-identifier searches found no earlier settlement of these four clauses in their stated scope. They do not establish global priority or exclude differently named results. The seventh clause’s source, proof and bounded literature limits are specified in Problems/oeis-a396798-seventh-iterate-mod-eight.md: the saved revision 12 was fetched with HTTP200, but the live refresh returned HTTP403. Exact Crossref A396798 returned HTTP200 with 0 results; the broad query’s first 20 unrelated titles are not exhaustive. arXiv returned HTTP429, and the code index’s zero missed known modules. These limits do not establish current live-page freshness or exclude differently indexed results.

Verified locator

• URL: https://oeis.org/A396798