Abstract: This paper investigates the deep structural relationships between the generalized Mersenne sequence $$H_n = 3^n - 1$$, the modular matrix $$T_c = \begin{pmatrix} c^2 + c + 1 & -c \\ c^2 & 1 - c \end{pmatrix} \in \Gamma_0(c^2)$$, and Fibonacci numbers. We show that iterating $$T_c$$ yields rational functions $$T_c^r(\infty) = \frac{P_r(c)}{Q_r(c)}$$ 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 $$H_n$$, 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 $$H_n = 3^n - 1$$.

Keywords: Mersenne numbers, Fibonacci numbers, modular group, Williams primes, Pascal’s triangle, recurrence relations

Pages: 8-19

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