That would pretty much mean a total break of SHA-256. The purpose of a cryptographically secure hash is to make the outputs computationally pseudorandom with respect to the inputs. You imply some clever way to use a correlation between the inputs and outputs for other types of calculations. Even MD5 (or even MD2) isn’t that broken.
There are ways to successively approximate PI using random numbers. Now, I'm not saying it *can* be done or even suggesting how (I'm smart but not that kind of smart). I'm just saying it wouldn't surprise me.