On a sequence related to Mersenne numbers and its connections to the Horadam sequence

Mersenne numbers, introduced by Marin Mersenne in the early 17th century and known since antiquity for their deep links to prime number theory, have long fascinated mathematicians. In this study, we introduce a second-order sequence derived from consecutive Mersenne numbers, which we refer to as the Trisenne sequence. Our objective is to investigate its structural properties and to establish relationships between the Trisenne numbers and the general Horadam (or generalized Fibonacci) sequence through their generating functions. The connection between these families is analyzed using both ordinary and exponential generating functions, with particular emphasis on classical cases such as the Fibonacci, Lucas, Pell, and Jacobsthal sequences. We also discuss the historical context and mathematical lineage of Horadam sequences, tracing their origin to the work of Edouard Lucas and Alwyn Horadam. Theoretical results are illustrated with explicit examples, and several related identities are presented.