R-sequencings and strong half-cycles from narcissistic terraces

We construct narcissistic terraces for cyclic groups that have various properties which enable the construction of R-sequencings and strong halfcycles for many non-cyclic abelian groups. Among other results, we show: that an abelian group which is isomorphic to a direct product of cyclic factors such that the number of factors of order congruent to 3 (mod 4) of order at most 79 is at least as large as the number of such factors of order greater than 79 is R-sequenceable (including all groups that are the direct product of cyclic groups of orders congruent to 1 (mod 4)); that for any abelian 3-group there are infinitely many R-sequenceable groups whose Sylow 3-subgroups are of that form; and that abelian groups whose Sylow 3-subgroups are of the form Z3×Z9×Z27×Z81 or Z3×Z9×Z27×Z81×Z3k where k ≡ ρ + σ (mod 2) are R-sequenceable. For strong half-cycles we give the first constructions for non-cyclic and non-elementary-abelian groups, including for groups that can be written as the product of cyclic factors, all either of order congruent to 1 (mod 12) or order at most 81 with order congruent to 1 (mod 4). We also show that for composite n with 21 ≤ n ≤ 69 there is a robust half-cycle for Zn.