OEIS A000013: Binary Necklaces

A deep dive into A000013, the classic two-color binary necklaces sequence. We’ll explore what it means to count distinct circular arrangements of n beads where rotations are considered the same, outline the jewel of the formula A(n) = (1/n) * Σ_{d|n} φ(d) * 2^{n/d} (with φ the Euler totient function), and unpack why divisors and totients appear. We'll illustrate with small n (n=1 → 2, n=2 → 3, etc.), connect the idea to shift registers in computer science (where cyclic binary outputs mirror the necklace counting), and glimpse how this elegant counting threads through combinatorics, coding theory, and symmetry concepts across math and CS.

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