OEIS A00018: Representations by the quadratic form x^2 + 16y^2

Today we tackle OEIS A00018, a sequence that counts how many positive integers up to 2^n can be written in the form x^2 + 16y^2. We unpack what that quadratic form means, why the search space doubles with each step, and how the first few terms—1, 1, 2, 2, 4, 8, 13, 25—hint at deeper number-theoretic structures. Along the way we connect this to Landau-type results on primes and to the ideas explored by Shanks and Schmidt, showing how a simple formula opens a window into the distribution of primes, quadratic forms, and the hidden geometry of numbers.

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