In previous papers, we defined functions f_m and c_m based on an arithmetical function f₀, and determined numbers of restricted words over a finite alphabet counted by these functions. In this paper, we examine the reverse problem: for each of the five specific types of restricted words, we find the initial function f₀ such that f_m and c_m enumerate these words. We derive explicit formulas for f_m and c_m.

Fibonacci, Mersenne, Pell, Jacosthal, Tribonacci, and Padovan numbers all appear as values of f_m. We derive their new combinatorial interpretations and the explicit formulas.