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I entered Rachel's continued fraction for Phi as 1+(1/(1+(1/(1+(1/(1+(1/(1+(1/(1+(1/(1+1)))))))))))) into Wolfram Alpha and got the fraction 34/21, two successive Fibonacci numbers, which I found very satisfying and exciting. Rachel's formulation had 13 "1"s, and 13 is the number before 21 in Fibonacci. The 13th "1" right at the bottom of Rachel's continued fraction pile has a plus sign and three dots after it, but to make my formulation suitable for Wolfram I put another "1" after that last plus sign instead, to seal it as it were rather than leave it open.
Anyway, I'm very interested in sequences similar to Fibonacci, especially for their potential to model natural structures and processes at least as well as Fibonacci's does, including those which differ minimally from the Fibonacci recurrence formula a(n) = a(n-1) + a(n-2). There's one known as Narayana's cows (OEIS A000930) whose recurrence has a "3" at the end instead of a "2", that is a(n) = a(n-1) + a(n-3) and has a ratio constant of roughly 1.4656, which I call the Bovine ratio, symbolised by the Greek(ish) letter Moo. Can anyone formulate the continued fraction for that ratio as Rachel did for Phi?