[seqfan] A proportion of odd digits vs the total

[Éric Angelini]

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 number of digits used in S 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?
Best,
É.