I think a standard reference for this is: D.H. Lehmer. Continued Fractions containing arithmetic progressions. Scripta Math. 29, 17--24, (1973). Doug Bowman
I recently asked, on one of these networks, about the number whose continued fraction is {1,2,3,4,...} and someone was kind enough to provide an explicit answer, which I've stupidly deleted. Can it be repeated?
Incidentally, is there a more general result, that a continued fraction whose partial quotients form an arithmetic progression (or a set of APs) can be expressed in terms of Bessel functions?
There are a few examples which are rational functions of e. R.
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