On representations of a number as a sum of three squares

We give a variety of results involving s(n), the number of representations of n as a sum of three squares. Using elementary techniques we prove that if 9 + n, s(9^λn) = (3^λ+1-1/2-(^-n/3)3^λ+1-1/2)s(n); via the theory of modular forms of half integer weight we prove the corresponding result with 3 replaced by p, an odd prime. This leads to a formula for s(n) in terms of s(n′), where n′ is the square-free part of n. We also find generating function formulae for various subsequences of {s(n)}, for instance Σs(3n + 2)qⁿ = 12 Π (1 + q^2n-1)²(1 - q⁶ⁿ)³. n≥0 n≥1