[seqfan] Balanced / Unbalanced numbers

[Éric Angelini]

Hello SeqFans,
let's call "Balanced" the integers of A036301 and "Unbalanced" the others.

[http://oeis.org/A036301 
"Numbers n such that sum of even digits of n equals sum of odd digits of n."]

Take an Unbalanced and add to it its closest Balanced; 
if the result is Balanced, stop. 
If the result is Unbalanced, iterate.

Question:
Do all Unbalanced end on a Balanced?

Example with the Unbalanced "1":
1 + 112 = 113
113 + 112 = 225
225 + 211 = 436
436 + 431 = 867
867 + 871 = 1738
1738 + 1744 = 3482
3482 + 3476 = 6958 is Balanced.

Example with the Unbalanced "2":
2 + 112 = 114
114 + 112 = 226
226 + 211 = 437
437 + 431 = 868
868 + 871 = 1739
1739 + 1744 = 3483
3483 + 3489 = 6972
6972 + 6963 = 13935
13935 + 14003 = 27938
27938 + 27968 = 55906
55906 + 55820 = 111726
111726 + 111728 = 223454
223454 + 223449 = 446903
446903 + 446905 = 893808
893808 + 893813 = 1787621 is Balanced.
(if I made no mistakes)

P.-S.
If there are two possible Balanced to be added to an Unbalanced (as this Unbalanced would stand at the same distance of the two Balanced) compute both branches.
Best,
É. at Brussels time and day 19:29 Dec 7, 2018