Array of recurrence "start with 2, square and add k" begins:
k..|.A[k,n]=A[k,n-1]^2 + k
-1.|.2..3...8....63.......3968..15745023.247905749270528.............A003096
0..|.2..4..16...256......65536..4294967296.18446744073709551616......A001146
1..|.2..5..26...677.....458330..210066388901.44127887745906175987802.A003095
2..|.2..6..38..1446....2090918.4371938082726...19113842599189892819591078...
3..|.2..7..52..2707....7327852.53697414933907..2883412370584178505178284652.
4..|.2..8..68..4628...21418388.458747344518548.210449126102819371741916028308.
5..|.2..9..86..7401...54774806.3000279372337641.9001676312074749038996905444886.
6..|.2.10.106.11242..126382570.1597255405035792810...
7..|.2.11.128.16391..268664888.72180822044052551...
8..|.2.12.152.23112..534164552.285331768613360712..
9..|.2.13.178.31693.1004446258.1008912285210202573.
10.|.2.14.206.42446.1801662926.3245989298922881486.
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