A proportion (odd digits vs all digits)

Obviously (and unfortunately) no one has been interested in this two week-old message on SeqFan:
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> Hello SeqFans, [posted from Brussels, Belgium, May 6th at 17:08]

We want a(n)/a(n+1) to represent a proportion where:
--> a(n) is the number of odd digits used in the seq up to a(n) -- a(n) included;
--> a(n+1) is the total number of digits used so far.

S = 1, 2, 3, 5, 8, 12, 19, 31, 49, 61, 72, 83, 94, 105, 116, 132, 200, ...

I think that this is true for the first 7 proportions here, but I am 100% sure that from a(7) on this is lexico-wrong (perhaps even earlier :-(

Let's see:
1/2 means that there is 1 odd digit in the first 2 digits of S 
      [true: they are respectively (1) and (1,2)]
2/3 means that there are 2 odd digits in the first 3 digits of S 
      [true: they are respectively (1,3) and (1,2,3)]
3/5 means that there are 3 odd digits in the first 5 digits of S 
    [true: they are respectively (1,3,5) and (1,2,3,5,8)]
5/8 means that there are 5 odd digits in the first 8 digits of S 
     [true: they are respectively (1,3,5,1,1) and (1,2,3,5,8,1,2,1)]
8/12 means that there are 8 odd digits in the first 12 digits of S 
     [true: they are respectively (1,3,5,1,1,9,3,1) and (1,2,3,5,8,1,2,1,9,3,1,4)]
12/19 means that there are 8 odd digits in the first 12 digits of S
     [true: they are respectively (1,3,5,1,1,9,3,1,9,1,7,3) and (1,2,3,5,8,1,2,1,9,3,1,4,9,6,1,7,2,8,3)]
19/31 means that there are 19 odd digits in the first 31 digits of S 
[true: they are respectively (1,3,5,1,1,9,3,1,9,1,7,3,9,1,5,1,1,1,3) and (1,2,3,5,8,1,2,1,9,3,1,4,9,6,1,7,2,8,3,9,4,1,0,5,1,1,6,1,3,2,2)
etc.
Could someone compute S, if this is of interest?
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14 May 2020 update
I think the hereunder new S beats the example above:

Sn = 1, 2, 3, 5, 8, 11, 21, 31, 41, 51, 62, 80, 201, 311, 331, 351, 371, 391, 511, 531, 551, 571, 590, 711, 731, 751, 761, 771, 781, 791, 801, 812, 822, 832, 842, 852, 862, 872, 882, 892, 902, 912, 922, 932, 942, 952, 962, 972, 982, 992, 1002, ...
Let's see now:
1/2 means that there is 1 odd digit in the first 2 digits of S 
      [true: they are respectively (1) and (1,2)]
2/3 means that there are 2 odd digits in the first 3 digits of S 
      [true: they are respectively (1,3) and (1,2,3)]
3/5 means that there are 3 odd digits in the first 5 digits of S 
    [true: they are respectively (1,3,5) and (1,2,3,5,8)]
5/8 means that there are 5 odd digits in the first 8 digits of S 
     [true: they are respectively (1,3,5,1,1) and (1,2,3,5,8,1,2,1)]
8/11 means that there are 8 odd digits in the first 11 digits of S 
     [true: they are respectively (1,3,5,1,1,1,3,1) and (1,2,3,5,8,1,1,2,1,3,1)]
11/21 means that there are 11 odd digits in the first 21 digits of S 
     [true: they are respectively (1,3,5,1,1,1,3,1,1,5,1) and (1,2,3,5,8,1,1,2,1,3,1,4,1,5,1,6,2,8,0,2,0)]
21/31 means that there are 21 odd digits in the first 31 digits of S 
     [true: they are respectively (1,3,5,1,1,1,3,1,1,5,1,1,3,1,1,3,3,1,3,5,1) and (1,2,3,5,8,1,1,2,1,3,1,4,1,5,1,6,2,8,0,2,0,1,3,1,1,3,3,1,3,5,1)]
31/41 means that there are 31 odd digits in the first 41 digits of S
     [true: they are respectively (1,3,5,1,1,1,3,1,1,5,1,1,3,1,1,3,3,1,3,5,1,3,7,1,3,9,1,5,1,1,5) and (1,2,3,5,8,1,1,2,1,3,1,4,1,5,1,6,2,8,0,2,0,1,3,1,1,3,3,1,3,5,1,3,7,1,3,9,1,5,1,1,5)]; etc.
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