OEIS A000144: Representations of n as the sum of 10 squares

In this episode we explore A000144—the number of representations of an integer n as a sum of 10 squares (ordered 10‑tuples of integers whose squares sum to n). For example, a(1) = 20 and a(2) = 180. We trace the mathematical lineage from quadratic forms and Lovell’s 19th‑century formulas to Ramanujan’s world of modular forms, explaining how these ideas illuminate why such representations exist and how they are counted. We’ll dive into the Euler transform method, showing how applying it to the repeating sequence 20, −30, 20, −10 (and continuing periodically) yields the terms of A000144, and we’ll connect this with the Jacobi theta function generating framework. The episode also highlights the interconnections with other OEIS sequences (A000456, A04068, A030212) and discusses the real‑world relevance of sums of squares in cryptography and coding theory. A000144 serves as a vivid example of how quadratic forms, modular forms, and generating functions weave together in the OEIS tapestry.

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