Sum three terms & digits

[Eric Angelini]

Hello Seq-Fans,

Is this sequence finite? I guess yes:

S = 1,2,3,4,5,7,8,9,10,14,15,31,32,33,34,35,37,38,39,40,42...

Sum three consecutive terms of S: the result cannot share
any single digit with any of the three considered terms.

Examples:

1+2+3=6 and "6" has no common digit with 1 or 2 or 3
8+9+10=27 and "27" has no common digit with 8 or 9 or 10

But:

9+10+11=30 and "30" shares the digit "0" with "10" -- thus
           "11" is not written after 10
9+10+12=31 and "31" shares the digit "1" with "10" (and "12")
           thus "12" is not written after 10
9+10+13=32 and "32" shares "3"... thus... no "13"

Now:

9+10+14=33 and "33" shares nothing, thus 33 is written after
           9,10, etc.

Best,
É.