I once read about an interesting formula/series for π that rather than converging on pi or some multiple or fraction of it, instead converged on the decimal portion of it, π minus the 3. Another part that I might be misremembering is how the first term had a 2,3, and 4 in it, and the next term had 4,5,6 and then 5,6,7 or some progression like that. Does this ring any bells with anyone? Does it have a name I can use to look up more info on why it works and how it weirdly seems base-10 centric?
Wolfram Mathworld has a number of different formula’s for pi, of which equation #29 sort of fits your requirement. It starts out with 3+… and then the rest and involve a number of terms including some consecutive integers.
Note that there is nothing in your description that is specific to base 10. Regardless of your base, the integer and fractional part of the number will be separated, and regardless of base, consecutive sets of three integers will continue to be consecutive. If you wrote all the numbers in binary or base 13 the formula would still work.
Equation 13 in that Wolfram link appears to be exactly what the OP is talking about. I agree it has nothing to do with base 10.
Thank you, 12 & 13 are almost certainly it. The one @Buck_Godot pointed to seemed way more complicated than I remembered.
However, I don’t recall anything about it summing to 1/4 of the decimal portion. I’m not too good with the math for series sums, is it possible to multiply by 4 on both sides to put 4 in the numerators of all the terms to get rid if the 1/4 and it would still work?
Thanks for correcting me about the base issue. Strike that part. 
The interesting thing to me is how pi-3 pops out of this, which by itself seems to be just another transcendental number that just so happens to be 3 away from pi.
Sure, you can do that. It’s just easier to write the 4 once, than to write it an infinite number of times, so it’s conventional to factor it out.
Well, if you don’t write the 4 an infinite number of times, you have to write 1 an infinite number of times, so it seems like mostly a wash.