Incidentally, I ran into the "Ramanujan Machine" and was wondering if the many formulas for certain constants can be generalized. For starters, what is the most simple nontrivial CF $1/(a_1+1/(a_2+1/(a_3+...)$ with $a_i=(a*i+b)/(c*i+d)$ with constant $a,b,c,d,ad-bc=1$? I googled a bit, but results are very scattered. (Some formulas with parameters by Zeilberger were in the NATURE article.) One might say I look for what Gradshteyn collected for integrals.
(addendum: If one can rely on Quora, not even the most simple $a_i=i$, numerically $1.433...$, is known, thus results seem to be scattered even generating-wise)
