Relations Among Hn, Tc and Fibonacci Sequences
PROOF, vol.6, pp.8-19, 2026 (Peer-Reviewed Journal)
- Publication Type: Article / Article
- Volume: 6
- Publication Date: 2026
- Doi Number: 10.37394/232020.2026.6.2
- Journal Name: PROOF
- Page Numbers: pp.8-19
- Karadeniz Technical University Affiliated: Yes
Abstract
This paper investigates the deep structural relationships between the generalized Mersenne sequence Hn = 3 n − 1, the modular matrix Tc = [ c 2 +c+1 −c c 2 1−c ] ∈ Γ0(c 2 ), and Fibonacci numbers. We show that iterating Tc yields rational functions T r c (∞) = Pr(c) Qr(c) whose coefficients appear in Pascal’s triangle and whose evaluations at c = 1 give Fibonacci numbers. We derive recurrence relations, matrix representations, and combinatorial identities for Hn, and explore its connections to Williams primes, Pell equations, modular forms, and various fields including number theory, linear algebra, and dynamical systems. The study uncovers unexpected links among these mathematical objects, highlighting the rich combinatorial and algebraic structure of the sequence Hn = 3 n −1.