Recurrence relations for the squares of the Horadam numbers and some associated consequences

Abstract

We derive recurrence relations for the squares of the Horadam numbers w2n, where the Horadam sequence wn is such that the numbers wn, for n ∈ Z, are defined recursively by w0 = a, w1 = b, wn = pwn−1 − qwn−2 (n ≥ 2), where a, b, p and q are arbitrary complex numbers with p ≥ 0 and q ≥ 0. Some related results emanating from the recurrence relations such as reciprocal sums, partial sums, and sums with double binomial coefficients are also presented.