Recurrence Relations for the Squares of the Horadam Numbers and Some Associated Consequences

We derive recurrence relations for the squares of the Horadam numbers wn2 w_n^2 , where the Horadam sequence w_n is such that the numbers w_n, for n ∈ ℤ, are defined recursively by w₀ = a, w₁ = b, w_n = pw_n−1 − qw_n−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.