Two-fold Langford sequences exist exactly when 3l ≥ 2d − 1; the m-fold counting bound and its attainment, and a residue bound
A Lean 4 characterization of two-fold Langford sequences. An m-fold Langford sequence of order l and defect d fills the positions 1, ..., 2ml so that every p in [d, d + l - 1] occupies the endpoints of m pairwise disjoint pairs of positions at distance p. For d, l >= 1 a two-fold sequence exists exactly when 3l >= 2d - 1. For every m, (2m - 1)l >= 2d - 1 is necessary, with l(2d + l + 1) = 0 (mod 4) for odd m, and the bound is attained whenever 2d + l = 2ml + 1; the row of order one exists exactly when d divides m; no three-fold sequence of order 3 and defect 6 exists; and for every m >= 3…
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