OEIS A000031: Number of binary necklaces with n beads counted up to rotation

We explore how a simple two-color necklace problem (no flips) captured by OEIS A000031 opens a gateway to rich mathematics: Euler’s totient, de Bruijn cycles, irreducible polynomials, shift registers, and finite fields. From combinatorics to algebra and cryptography, we see how counting necklaces links patterns, number theory, and real-world computing.

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