The theory of quasimodular forms is used to evaluate the convolution sums [Formula: see text] for all positive integers n. As a consequence, the number of representations of a positive integer n by the octonary quadratic form [Formula: see text] is determined.
For any positive integer n, we state and prove formulas for the number of solutions in integers of x 2 + y 2 + 5z 2 = n 2 , x 2 + y 2 + 6z 2 = n 2 and x 2 + 2y 2 + 3z 2 = n 2 , which were first conjectured by S. Cooper and H. Y. Lam.
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