ABSTRACT 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 .
Zhang et al. (Sun,) studied this question.
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