OEIS A000122: Sum of Two Squares and the Jacobi Theta Connection

A000122 counts the ways to write n as a sum of two squares; its coefficients appear in the expansion of the Jacobi theta function theta_3. We explain the zeros-and-twos pattern (twos at square indices), explore the link to r2(n), and trace a rich web of connections to Ramanujan theta functions, Dirichlet convolution, lattices, and modular forms. Along the way we’ll discuss asymptotic mean 2π/5 and point you to the OEIS entry for formulas, references, and further links.

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