[seqfan] Sum of all valid substrings (VS)

[Eric Angelini]

Hello SeqFans,

A004207...  
http://www.research.att.com/~njas/sequences/A004207
... goes like this:

    a(1) = 1, a(n) = sum of digits of all previous terms. 
    1, 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101,
      103, 107, 115, 122, ...

    Comment: If the initial 1 is dropped, we get the se-
    quence defined by a(1) = 1, a(n) = a(n-1) + sum of di-
    gits of a(n-1).
    (...)
-----
Let's try to build another sequence, slightly changing this:
     a(n) = a(n-1) + sum of digits of a(n-1)
into a(n) = a(n-1) + sum of all valid substrings of a(n-1)

What would be a valid substring (VS), here?

123 has VS 1, 2, 3, 12 and 23
1234 has VS 1, 2, 3, 12, 23, 34, 123, 234
12045 has VS 1, 2, 0, 4, 5, 12, 20, 45, 120, 204, 1204 and 2045
etc.

Where would one go starting with 1? Any loops, fixed points,
ups and downs?

Best,
É.