The Integer Source and Dyadic Parity of OEIS A397592

Abstract

Integer existence and uniqueness for the literal source of A397592, together with its dyadic-neighbor parity assertion for every source solution.

The source predicate is D5/S1/Recurrence/LinearRows/DoubledLinearExponentDyadicSupport.SourceA397592. For A in Z[[X]], A_Q denotes A.map (Int.castRingHom Rat), its coefficientwise integer-to-rational image in Q[[X]]: coeff j(A_Q) is the rational image of coeff j(A) for every natural j. The sequence starts at index zero, with a(n)=coeff n(A) and a(0)=1. The operation rescale(1/m,A_Q) means A_Q(X/m). Its mth power is ordinary multiplication of formal series, with no factorial normalization or convergence premise.

Definition 1.1 (The literal rational-rescaling source).

$$\forall A:\mathrm{PowerSeries}\left(\mathbb{Z}\right),SourceA397592\left(A\right)⟺(\mathrm{constantCoeff}\left(A\right)=1\wedge (\forall m:\mathbb{N},0<m⟹\sum _{j=0}^{m-1}\mathrm{coeff}\left(j,(\mathrm{rescale}\left(\frac{1}{m},A_{\mathbb{Q}}\right){)}^m\right)=2^{m-1}))$$

Formalization. D5/S1/Recurrence/LinearRows/DoubledLinearExponentDyadicSupport.SourceA397592 (✓ std3).

Citation. Paul D. Hanna (2026). OEIS A397592: doubled linear rows and dyadic parity. URL: https://oeis.org/A397592.

Commentary.

The sum contains exactly the first m coefficients, at indices j=0 through m-1, for every natural m>0. The equality is in Q and the unknown series has integer coefficients. This is the NAME equation with the constant term specified by the source’s ordinary generating series and offset zero.

Theorem 1.2 (Integer well-posedness and Hanna’s parity assertion).

$$(\exists !A:\mathrm{PowerSeries}\left(\mathbb{Z}\right),SourceA397592\left(A\right))\wedge (\forall A:\mathrm{PowerSeries}\left(\mathbb{Z}\right),SourceA397592\left(A\right)⟹\forall n:\mathbb{N},3<n⟹(\mathrm{Odd}\left(\mathrm{coeff}\left(n,A\right)\right)⟺(\exists k:\mathbb{N},1<k\wedge (n=2^k-1\vee n=2^k+1))))$$

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

Resolves. Problems/oeis-a397592-dyadic-parity (proved) by D5/S1/Recurrence/LinearRows/DoubledLinearExponentDyadicSupport.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397592: doubled linear rows and dyadic parity. URL: https://oeis.org/A397592.

Commentary.

The first conjunct asserts existence of an integer source and uniqueness among all integer series satisfying the literal equation. The second applies to every such series and every natural n>3. Oddness is equivalent in both directions to one of the two indices 2^k-1 and 2^k+1 for a natural k>1. The A397591 series starts at index one with zero constant term and has a different defining equation; it is not the source named here.

References

• Truth anchor: D5/S1/Recurrence/LinearRows/DoubledLinearExponentDyadicSupport.SourceA397592
• Truth anchor: D5/S1/Recurrence/LinearRows/DoubledLinearExponentDyadicSupport.SourceA397592
• Truth anchor: D5/S1/Recurrence/LinearRows/DoubledLinearExponentDyadicSupport.result
• Dependency: D5/S1/Recurrence/LinearRows/LinearExponentDyadicSupport