OEIS A000086: Number of solutions to x^2+x+1 ≡ 0 (mod n)

We explore A000086, the function counting residues x that satisfy x^2+x+1 ≡ 0 (mod n). We’ll unpack how this congruence behaves, including its equivalence (via a shift) to x^2+1 ≡ 0, its multiplicative nature, and the prime-power rules: for p=3, e=1 yields 1, while higher powers give 0; for primes p ≡ 1 (mod 3) the value is 2; for primes p ≡ 2 (mod 3) it’s 0. The pattern of 0 or powers of 2 hints at underlying field extensions. We’ll also touch on connections to totient-related sequences modulo 3, the asymptotic mean around 0.367552, and links to polygonal numbers and sublattice enumeration—showing how a simple modular question opens doors to rich algebraic structures.

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