Continuing the discussion from the previous post, Ramanujan also gave a continued fraction for as,

where the , starting with *n* = 1, are given by the *linear* function,

(Notice the difference from the other version since this one has the cubes twice used as numerators.) Using a similar approach to Apery’s of finding a faster converging version, I found via *Mathematica* that,

where the are now given by the *cubic* function,

Of course, a more rigorous mathematical proof is needed that indeed the equality holds.

**Update**: J.M. from mathstackexchange.com proved that the continued fraction DOES converge to by connecting it to Apery’s version! One consequence of his analysis is that Apery’s generating polynomial can be seen as,

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