Pythagorean triangles with legs less than n

We obtain asymptotic estimates for the number of Pythagorean triples (a,b,c) such that a<n, b<n. These estimates (considering the triple (a,b,c) different from (b,a,c)) is (4π−2 log(1+2))n+O(n) in the case of primitive triples, and (4π−2 log(1+2))n log n+O(n) in the case of general triples. Furthermore, we derive, by a self-contained elementary argument, a version of the first formula which is weaker only by a log-factor. Also, we tabulate the number of primitive Pythagorean triples with both legs less than n, for selected values of n⩽1 000 000 000, showing the excellent precision obtained.