Hanna’s Quadrupled Linear Rows and Dyadic Parity

Abstract

An integer source series exists for OEIS A397594, and every source series has Hanna’s dyadic-neighbor odd support.

The ordinary formal series has a(n)=PowerSeries.coeff n A, with index zero and a(0)=1. Rational rescaling substitutes X/m after mapping each integer coefficient to Q. All powers are formal-series products. Subtraction between natural indices is truncated natural subtraction; no analytic convergence is required.

Definition 1.1 (The rational-rescaling source equation).

$$\forall A\in \mathrm{PowerSeries}\left(\mathbb{Z}\right),\mathrm{IsSourceSeries}\left(A\right)\iff ((\mathrm{PowerSeries}.\mathrm{constantCoeff}\left(A\right)=1)\wedge (\forall m\in \mathbb{N},(0<m)\Rightarrow (\sum _{j\in \mathrm{Finset}.\mathrm{range}\left(m\right)}\mathrm{PowerSeries}.\mathrm{coeff}\left(j,(\mathrm{PowerSeries}.\mathrm{rescale}\left((m:\mathbb{Q}{)}^{-1},\mathrm{PowerSeries}.\mathrm{map}\left(\mathrm{Int}.\mathrm{castRingHom}\left(\mathbb{Q}\right),A\right)\right){)}^m\right)=(4:\mathbb{Q}{)}^{m-1})))$$

Formalization. D5/S1/Recurrence/LinearRows/QuadrupledLinearExponentDyadicSupport.IsSourceSeries (✓ std3).

Citation. Paul D. Hanna (2026). OEIS A397594: quadrupled linear rows and dyadic parity. URL: https://oeis.org/A397594.

Commentary.

OEIS A397594 revision 15, NAME and FORMULA (1), without pagination: “G.f. A(x) such that the sum of the first n coefficients in A(x/n)^n equals 4^(n-1) for n >= 1.” “(1) 4^(n-1) = Sum_{k=0..n-1} [x^k] A(x/n)^n for n >= 1.” The natural m below is the source’s n, and j is its k. Finset.range m contains exactly j=0 through m-1. The equation is in Q, with A in PowerSeries Z and constant coefficient one.

Definition 1.2 (Integer existence and the parity assertion).

$$claim\iff ((\exists A\in \mathrm{PowerSeries}\left(\mathbb{Z}\right),\mathrm{IsSourceSeries}\left(A\right))\wedge (\forall A\in \mathrm{PowerSeries}\left(\mathbb{Z}\right),(\mathrm{IsSourceSeries}\left(A\right))\Rightarrow (\forall n\in \mathbb{N},(3<n)\Rightarrow (\mathrm{Odd}\left(\mathrm{PowerSeries}.\mathrm{coeff}\left(n,A\right)\right)\iff (\exists k\in \mathbb{N},(1<k)\wedge (n=2^k-1\vee n=2^k+1))))))$$

Formalization. D5/S1/Recurrence/LinearRows/QuadrupledLinearExponentDyadicSupport.claim (✓ std3).

Citation. Paul D. Hanna (2026). OEIS A397594: quadrupled linear rows and dyadic parity. URL: https://oeis.org/A397594.

Commentary.

OEIS A397594 revision 15, COMMENTS, without pagination: “Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).” Here a(n)=PowerSeries.coeff n A, n and k are natural numbers, and Odd applies to an integer coefficient. Existence of an integer source series is included, and the equivalence quantifies over every source series.

Theorem 1.3 (Hanna’s parity assertion proved).

$$(\exists A\in \mathrm{PowerSeries}\left(\mathbb{Z}\right),\mathrm{IsSourceSeries}\left(A\right))\wedge (\forall A\in \mathrm{PowerSeries}\left(\mathbb{Z}\right),(\mathrm{IsSourceSeries}\left(A\right))\Rightarrow (\forall n\in \mathbb{N},(3<n)\Rightarrow (\mathrm{Odd}\left(\mathrm{PowerSeries}.\mathrm{coeff}\left(n,A\right)\right)\iff (\exists k\in \mathbb{N},(1<k)\wedge (n=2^k-1\vee n=2^k+1)))))$$

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

Resolves. Problems/hanna397594 (proved) by D5/S1/Recurrence/LinearRows/QuadrupledLinearExponentDyadicSupport.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397594: quadrupled linear rows and dyadic parity. URL: https://oeis.org/A397594.

Commentary.

Clearing the rescaling denominators gives the integer row equation. Triangular normalization has forcing 4^n(n+1)^(n-1) in positive degree. Coefficientwise iteration constructs an integer solution. The forcing vanishes modulo two, so normalized uniqueness and the homogeneous dyadic-support theorem give the parity equivalence for every source solution. The assertion concerns the parameter four; the entire parameter triangle and the source’s reversion identities are separate questions.

References

• Truth anchor: D5/S1/Recurrence/LinearRows/QuadrupledLinearExponentDyadicSupport.IsSourceSeries
• Truth anchor: D5/S1/Recurrence/LinearRows/QuadrupledLinearExponentDyadicSupport.claim
• Truth anchor: D5/S1/Recurrence/LinearRows/QuadrupledLinearExponentDyadicSupport.result
• Dependency: D5/S1/Recurrence/Invariants/PowerSeriesNormalizedContraction
• Dependency: D5/S1/Recurrence/Invariants/UnitReversionSquareParity
• Dependency: D5/S1/Recurrence/LinearRows/LinearExponentDyadicSupport