Does anyone know the exact value for the continued fraction of $$1+\cfrac{1}{3+\cfrac{3}{5+\cfrac{5}{7+\cfrac{7}{9+\ddots}}}}?$$
I already know that $$1+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{9\ddots}}}}=\frac{e^2+1}{e^2-1},$$ but I only figured that out by typing the decimal approximation into google of the first few terms of the continued fraction (before I knew the exact value) which took me to a math paper saying that $\frac{e^2+1}{e^2-1}$ roughly equals the decimal approximation I typed in. I then typed in the continued fraction of $\frac{e^2+1}{e^2-1}$ into wolfram alpha and it spat out $$1+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{9\ddots}}}}.$$ I have no idea how to solve these so please don't downvote, I'm just doing this in case it's useful to someone one day, and out of curiosity of course.