Weighted sums of squares via generalized eulerian polynomials

We give explicit formulas for the weighted sum of squares σm-1 j=0 zj (aj + b)2, where a, b ∈ C are given, and z ∈ C, z ≠ 0, 1 is the weight. In the case a, b ∈ Z and z ∈ Q, we show that there is a one-To-one correspondence between our weighted sums and Primitive Pythagorean Triples. The main tools we use are the Z-Transform of sequences and a generalization of Eulerian polynomials. Copyright © 2017 The Fibonacci Association. All rights reserved.