OEIS A000132: Five-square representations and theta series

In this Deep Dive, we explore A000132, the number of ways to write an integer as the sum of five squares. We’ll unpack the distinctive unit‑digit pattern modulo 5, explain why certain residues occur only when the original number is five times a square, and survey the main tools used to study the sequence: generating functions, Seiichi Maniyama’s recursive formula, and Daniel Sutil’s link to A000118 (the four-squares sequence). We also venture into the deeper connections: A000132 as the theta series of the five‑dimensional lattice Z^5, its relation to quadratic forms, and the footholds it provides into modular forms. With a visual graph from OEIS, we’ll see growth patterns and structure, offering a window into how a counting sequence unfolds into rich, interconnected mathematics.

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