Or this one equals the golden ratio:
This one is pi-3:
This one is just equivalent to the usual sigma notation for adding over an index:
Anywhere you see a ... you just recurse using the index, thus it handles the job of sigma and pi, as well as the continued fractions examples. When the index runs out it just fills in the expression as it would be without the ... For example this gives 1.6:
Even with this some continued fractions would be cumbersome, so you can combine with Lagrange's notation:
I thought of using the colon to indicate that after recursing the specified number of times the last ... is replaced with a specific value.