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

We derive recurrence relations for the squares of the Horadam numbers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mrow> <m:mi>w</m:mi> </m:mrow> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> </m:math> w_n^2 , where the Horadam sequence w n is such that the numbers w n , for n ∈ ℤ, are defined recursively by w 0 = a , w 1 = 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.