OEIS A000152: Number of representations of n as a sum of 16 squares

We dive into A000152, the sequence counting how many ways an integer can be written as the sum of 16 squares. From the first terms (1, 32, 480, 4480, 29152) to the generating function built from the Jacobi theta function, and a recursive formula linking to a186690, we explore how the sequence encodes rich number-theoretic structure. We also connect to the broader theory of sums of squares via the Wikipedia article, contrasting this 16-square case with Legendre's three-square and Lagrange's four-square theorems, and discuss why higher-square representations open up new patterns and questions.

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