[seqfan] Two easy sequences not in the OEIS

[Jeremy Gardiner]

I was surprised not to find these sequences in the OEIS:

1, 11, 12, 121, 122, 123, 1231, 1232, 1233, 1234, 12341, 12342, 12343,
12344, 12345, ...

1, 11, 21, 121, 221, 321, 1321, 2321, 3321, 4321, 14321, 24321, 34321,
44321, 54321, ...

In each case, the sum of digits of a(n) = n.

These might be extended beyond 123456789 and 987654321 as follows:

1987654321, 11987654321, 21987654321, 121987654321, 221987654321,
321987654321, ...

I'll be happy to submit these, if anyone thinks they are interesting?

Further, summing the digits of terms of both sequences:

2, 22, 33, 242, 343, 444, 2552, 3553, 4554, 5555, 26662, 36663, 46664,
56665, 66666, ... 

Sum of digits of a(n) = 2, 4, 6, 8, 10, 12, ...

However there is a difficulty extending this sequence beyond
12345678, 87654321 => 99999999

Any suggestions?

Jeremy