I was also taking a look at this:
>When does c first appear at a[t]?
For our a=7, b=37 and I assumed that 259 (c) appears at t where a[t] = 114 * 259.
I was surprised when I found out that it was 78 * 259 which is from the record of a=13, b=259.
The funny thing about that record, compared to our record is this:
Record for (1, c) ={3:114:16:15:1:259}
Record for (13, c) ={3:78:58:45:13:259}
{3:78:58:45:13:259}
{3:114:16:15:1:259}
The difference here is interesting (but I'm filing it under a fluke for now)
114 - 78 = 36 =6*6 (6 is the n we want)
58 - 16 =42 => 7*6 (the an)
45 - 15 =30 => 7*5 (the a(n-1))
Again, though. Could easily be a fluke.