Number of subsets of Z/nZ closed under negation and doubling: terms to n = 200 and program

a(n) is the number of subsets S of Z/nZ such that S = −S and 2S ⊆ S — equivalently, the number of subsets of the levels {0, …, ⌊n/2⌋} closed under j ↦ min(2j mod n, n − 2j mod n). First terms: 2, 3, 4, 4, 4, 9, 4, 7, 8, 15, 4, 24, 4, 27, 16, 28, 8, 81, 4, 104, … For odd n, a(n) = 2^k with k the number of orbits of ⟨x ↦ 2x, x ↦ −x⟩ on Z/nZ; for even n the functional graph carries trees rooted at 0 and, when 3 | n, at n/3, and is a forest exactly when n = 2^a or 3·2^a.

Files. b400466.txt: n and a(n) for n = 1..200 (OEIS b-file format). a400466.py: the tree formula over the functional graph, a brute-force check for n ≤ 20, and the b-file generator.

Origin. The sequence counts the sub-vocabularies of the axis construction of a 12 × 12 Jacquard pattern system, where the doubling of a line's offset from the centre is the generating operation (companion notes, doi:10.5281/zenodo.22965031; One Object, Three Descriptions, doi:10.5281/zenodo.22986529); there n = 12.

Licences. Data: CC BY 4.0. Code: AGPL v3.