[seqfan] From N to a palindrome by adding

[Eric Angelini]

Hello SeqFans,
Start from any integer N and try to reach
a palindromic number in as few steps as possible.
A step is an addition -- you add to N one of N's
digits. And you iterate from there.
I've computed what I think are the shortest (?)
paths for 1 to 25 (in a condensed
way).

[Some possible sequences after the list,
if the idea is not old hat.]

1+1=2
2+2=4
3+3=6
4+4=8
5+5=10+1=11
6+6=12+1=13+3=16+6=22
7+7=14+1=15+1=16+6=22
8+8=16+6=22
9+9=18+1=19+1=20+2=22
10+1=11
11+1=12+1=13+3=16+6=22
12+1=13+3=16+6=22
13+3=16+6=22
14+1=15+1=16+6=2
15+1=16+6=22
15+5=20+2=22
16+6=22
17+7=24+4=28+2=30+3=33
18+1=19+1=20+2=22
19+1=20+2=22
20+2=22
21+1=22
22+2=24+4=28+2=30+3=33
23+2=25+5=30+3=33
24+4=28+2=30+3=33
25+5=30+3=33
...

Questions:
1) how many steps does 2014 need to reach
a palindrome?
2) is there a general formula?
3) some sequences might be of interest
for the OEIS:
- smallest integer reaching a palindrome 
in n steps;
- smallest integer reaching a palindrome
in n different ways (15 is an example above);
...
Best,
É.