What question did this study set out to answer?

The aim is to explore the modular periodicity of the sequence T_k = 6·2^k + 1 and its fractal properties.

March 16, 2026Open Access

Modular Period Structure and Fractal Laws of the Affine Sequence Tₖ = 6·2ᵏ + 1

Key Points

The aim is to explore the modular periodicity of the sequence T_k = 6·2^k + 1 and its fractal properties.
Defined the OMARWA period for the sequence T_k mod m.
Demonstrated the Core Reduction Theorem for establishing periods.
Verified findings using Lean 4 with Mathlib.
Identified a super-period L = 30 for T_k via three independent methods.
Established that P(m) aligns with the order of 2 under modified conditions based on m.
Noted connections with exceptional Coxeter numbers.

Abstract

We study the integer sequence Tₖ = 6·2ᵏ + 1 (OEIS A004119) and establish a complete theory of its modular periodicity. For each positive integer m, we define the OMARWA period P (m) as the minimal period of the sequence Tₖ mod m, and prove the Core Reduction Theorem: P (m) = ord₌' (2) where m' = m/gcd (m, 6). We derive fractal period laws, a super-period L = 30 via three independent paths, and observe coincidences with exceptional Lie-theoretic Coxeter numbers. All principal theorems are formally verified in Lean 4 with Mathlib (654 declarations, zero sorry axioms, 25 modules). Software DOI: 10. 5281/zenodo. 19024222.

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