OEIS A000164: Sum of three squares

We explore A000164, the number of representations of n as a sum of three squares (with zero allowed) under the convention i ≥ j ≥ k ≥ 0. We discuss Legendre’s three-square theorem, which n fail to be written as three squares, plus practical computation via the Ant King–Somos formulas and generating functions, and how this fits into the broader landscape of representations by sums of squares, including connections to Lagrange’s four-square theorem and cryptographic applications.

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