OEIS A000048: Two-color binary necklaces with primitive period under rotation and color-swapping

A000048 counts binary necklaces of length n with two colors, where the pattern has primitive period n (no smaller repeating block), we consider rotations equivalent but not reflections (fixed orientation), and swapping the two colors yields the same necklace. In other words, we count aperiodic binary necklaces up to rotation with color interchange. This sequence also arises in several other areas: it equals the number of binary irreducible polynomials of degree n over GF(2) with trace 1; it ties to binary irreducible self-reciprocal polynomials under certain conditions; and it appears in coding theory (Varshamov–Tenengolts codes) and Boolean automata networks. For a small n, like n=5, there are 3 distinct necklaces (up to rotation and color swap): 00001, 00111, and 01011. This single sequence serves as a bridge between combinatorics, finite fields, and the dynamics of complex systems.

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