Crystal, break, crystal: the 1-additive sequences U(u, v) with odd u and Finch's Conjectures 7 and 8

Once its even terms are known, the odd part of a 1-additive sequence follows a recurrence of "exactly one" type, and each new even term adds a shift to it. We call a purely periodic solution of such a recurrence a crystal. For U(7, v), v even and 7 not dividing v, we show that the odd part begins on a crystal of period 32v − 7; the last even term 14v + 28 adds a shift to the recurrence, and the odd part first leaves the crystal at the odd number 14v + 35; after a transient of 615v + 753 steps (for v ≥ 38) it settles on a second crystal of period 56(v + 2). As a consequence we obtain Finch's 1992 Conjectures 7 and 8 for u = 7: U(7, v) has exactly v/2 + 2 even terms for all v ≥ 8, and its fundamental difference and period are D = 112(v + 2) and N = 6v + 10 for v ≥ 38. Previously, regularity was known for six values of v, and a proof of the formula for D under the assumption on the even terms had been announced by Cassaigne. The proofs are computer-assisted: the odd part, transient included, consists of 2333 runs whose endpoints are linear in v, and a finite criterion is verified with v as a parameter for v ≥ 440, with a separate certificate checker and an independent implementation, and directly for 8 ≤ v ≤ 438. The first two stages hold in the same way for U(u, v) with u = 9, 11, 13, and for u = 9 and u = 11 also the third stage is proved, including the least period of the second crystal, which gives Finch's Conjecture 7 and the formulas D = 144(v + 2) and D = 704(v + 2) for these u. In a finite scan of 284 pairs with 7 ≤ u ≤ 63, a period from the first index was detected exactly for the pairs with u ≡ 1 (mod 4), u = 2^k − 1 or u = 11. The proofs are written out in full and the tables, programs and logs are provided, but their validity is left open to external verification; see the note at the end of the paper. The accompanying archive contains the block tables, the programs for the symbolic and direct verifications with their certificates and separate checkers, the independent implementation with its report, and the program for the scan.