Array of repeated conversions from decimal to base k

[Jonathan Post]

I don't see evidence of OEIS of this array, which unifies at least 9
existing sequences.

Recursively: T(k,1) = 10; T(k,n+1) = T(k,n) converted to base 10 from base
k, written in base 10.

The table begins as (row is what A-number):

1 | 10, 1111111111, ...
2 | 10, 1010, 1111110010, ... (A006937)
3 | 10, 101, 10202, 111222212, ... (A023372)
4 | 10, 22, 112, 1300, 110110, ... (A023378)
5 | 10, 20, 40, 130, 1010, 13020, ... (A023383)
6 | 10, 14, 22, 34, 54, 130, 334, 1314, ... (A023387)
7 | 10, 13, 16, 22, 31, 43, 61, 115, ... (A023390)
8 | 10, 12, 14, 16, 20, 24, 30, 36, 44, 54, ... (A008557)
9 | 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 22, ... (A023392)
10| 10, 10, 10, 10, 10, 10, 10, 10, ... (A010692)

This table can be presented by antidiagonals.  It has finite such
presentation, as we hit nondecimal characters. (exercise for reader: when?).

The main diagonal is: T(n,n) = 10, 1010, 10202, 1300, 1010, 130, 61, 36, 18,
10, ...

This is not itself "base" as it explicitly deals with conversions
nonarbitrarily.  It may be "dumb."  But it does, as I say, unify at least 9
existing sequences.

See also A063432.

Happy holidays,

-- Jonathan Vos Post
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