Generalized Signed Langford Sequences

This paper introduces an extension of signed Langford sequences to generalized signed Langford sequences with three distinct types and investigates their existence. We establish three main existence results: (1) There exists a generalized signed Langford sequence of Type 3 except for (a) and , (b) and . (2) For all admissible positive integers and , there exist two disjoint generalized signed Langford sequences of both Type 1 and Type 2. (3) For parameters satisfying with , and with , there exist two disjoint generalized signed Langford sequences of Type 3. Additionally, we prove the existence of absolutely symmetric signed Langford sequences in four special cases: when with , and when with .