Computations on the 1-additive sequences U(4, v), v ≡ 3 (mod 4)

Finch conjectured that for odd v ≥ 5 the 1-additive sequence U(4, v) has only the even terms 4, 2v + 4, 4v + 4 (and one more when v = 2^m − 1), which implies that its differences are eventually periodic; this was proved by Cassaigne and Finch for v ≡ 1 (mod 4) and, by a finite computation, by Hinman, Kuca, Schlesinger and Sheydvasser for v = 11, 19. We report computations for v ≡ 3 (mod 4). For every such v with 7 ≤ v ≤ 107 other than 15, 31, 63 we verify the conjecture, by checking the finite criterion of Hinman et al. (for v ≤ 91) or a direct variant of it (for v = 95, 99, 103, 107), and list the period N and the fundamental difference D; the values for 67 ≤ v ≤ 107 appear to be new, and for v = 107 the periodic part starts only after about 2.3·10^13 odd numbers. For U(4, 15), whose period is an open problem of Finch, we found no further even term below 4.343·10^13 and detected no period within 2.17·10^13 steps of the recurrence, which rigorously excludes periods of length at most 2^16 starting within 2·10^13 steps; a heuristic extrapolation from the irregular cases puts transient plus period around 10^29. The same direct variant shows that U(6, b) is regular for b = 13, 17, 19, 23; for b = 13, 19 the fundamental difference is 2^7·b, and for b = 17, 23 the periods have no such structure, with D ≈ 9.4·10^7 and 4.7·10^11. Finally, for v ≡ 3 (mod 8) and v ≡ 7 (mod 16) we show, by a computer-assisted symbolic verification, that the recurrence governing the odd terms has a periodic solution of least period 2^(m+2)(v + 1), 2^m < v < 2^(m+1), of the same width as in the Cassaigne–Finch formula; this concerns the recurrence, not the sequences themselves. All results depend on computer calculations; see the note at the end. The accompanying archive contains the programs, the recorded outputs and the saved states of the long runs.