OEIS A000205: Representable integers by x^2 + 3y^2 up to 2n

An exploration of A000205—the count of positive integers ≤ 2n that can be written as x^2 + 3y^2. We unpack the binary quadratic form x^2 + 3y^2, its discriminant -12, and why representability ties to modular criteria and prime-factor parity via Thue–Vinogradov methods. In particular, n is representable by x^2 + 3y^2 iff the exponent of 2 in n or 2n is even and every prime p ≡ 2 (mod 3) occurs to an even power; a window into the wider theory of quadratic forms and class numbers.

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