OEIS A000143: Sums of eight squares

In this episode we dive into A000143, the count of representations of a nonnegative integer as a sum of eight squares when both order and signs matter. We start with simple cases (a(0)=1, a(1)=16, a(2)=112) and explore the elegant structure behind the sequence: a(n) = 16·b(n) where b(n) is multiplicative; explicit prime-power formulas exist for b(p^e) (different treatments for odd primes and for p=2); a convenient divisor-sum expression uses sums over divisors with cubed terms, and there is a generating function given by the Jacobi theta function theta3(q^8). We also touch on the deeper connections to modular forms and theta functions, and what these patterns reveal about representations of numbers as sums of eight squares.

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