On Weighted Sums of Horadam Numbers

For nonzero integers ????, ???? with ????-???? ≠ 1, let (???? ????) ????≥0 be the Lucas sequence of the first kind defined by ???? ???? = ???????? ????-1-???????? ????-2 for ???? ≥ 2 with initial terms ???? 0 = 0 and ???? 1 = 1. In this paper, we provide explicit formulas for evaluating the weighted sums ∑ ???? ???? ???? ????=1 ???? ???? and ∑ ???? ???? ???? ????=1 ???? ????+???? using binomial coefficients for nonnegative integers ???? and arbitrary integers ????. Furthermore, we extend these summation formulas to the Horadam sequence (???? ????) ????≥0 defined by the same recursion ???? ???? = ???????? ????-1-???????? ????-2 for ???? ≥ 2 with arbitrary initial terms ???? 0 and ???? 1 .