The main object of the present paper is to reveal connections between Mersenne numbers Mₙ=2ⁿ-1 and generalized Fibonacci (i.e., Horadam) numbers wₙ defined by a second order linear recurrence wₙ=pwₙ₋₁+qwₙ₋₂, n≥ 2, with w₀=a and w₁=b, where a, b, $p>0$ and q≠0 are integers. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. Several explicit examples involving Fibonacci, Lucas, Pell, Jacobsthal and balancing numbers are stated to highlight the results.
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Frontczak et al. (2020) studied this question.
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