A new parameterisation of Pythagorean triples in terms of odd and even series

In this paper we introduce a formula that parameterises the Pythagorean triples as elements of two series. With respect to the standard Euclidean formula, this parameterisation does not generate the Pythagorean triples where the elements of the triple are all divisible by 2. A necessary and sufficient condition is also proposed for a Pythagorean triple obtained from this formula to be primitive.