Fibonacci and Mersenne Sums: A Reversible Path Through Periods 60–12–3–3

This paper presents a numerical experiment in which the Fibonacci sequence F_n is added to the Mersenne numbers M_n = 2^n - 1, with a shift of one index. The resulting sequence A_n = F_n + Mₙ₊₁ is then subjected to three successive applications of the neighbour-sum transform T, defined as (T a)_n = aₙ₋₁ + aₙ₊₁. This process yields a cascade of tail periods: 60, 12, 3, and 3. We show that these periods are masks hiding the classical Fibonacci and Lucas sequences. The masks are removed by simple operations: first differences, second differences, and division by 5. Explicit formulas for the recovery are given and verified numerically. All transformations are reversible, confirming that the cascade 4 -> 20 -> 4 -> 60 -> 12 -> 3 -> 3 -> ... is universal and robust under a wide class of additive perturbations, including Mersenne numbers. This experiment complements previous studies using constant shifts and triangular numbers.