OEIS A00026: Even Sequences with Period 2n

We explore A00026, counting the 'even' orbits of binary necklaces of length 2n under the combined dihedral symmetry DN cross S2 (rotations, reflections, and color inversion). A binary necklace is a binary sequence up to rotation and flip; here inversion flips 0s and 1s. The 'even' condition selects a special subset whose cardinality is obtained via Burnside's lemma. The resulting count depends on the parity of n and relates to the ordinary binary-necklace counts (A000011); we'll unpack the group action, fixed points, and how the period-2n constraint shapes the orbit structure.

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