User victor p - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:59:17Z http://mathoverflow.net/feeds/user/30747 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119241/harmonacci-recurrence-and-identities-for-pi "Harmonacci" recurrence and identities for $\pi$ Victor P 2013-01-18T07:11:49Z 2013-01-18T11:57:45Z <p>While playing with something totally irrelevant I stumbled upon the recurrence: $$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$</p> <p>It turns out that given $a_0 = 1, a_1 = 1$,</p> <p>$$lim \frac{a_{2n}}{a_{2n-1}} = \frac{\pi}{2}$$</p> <p>I have a very crude idea (or rather a hint) on proving it (the iterations sort of unfold into a sort of Viete product, which is sort of expected), but my technique is rusty at best.</p> <p>With different initial conditions, things start getting really scary, for example $a_0 = 2, 3, 4, 5$ yield $\frac{8}{\pi}, \frac{9\pi}{8}, \frac{128}{9\pi}, \frac{225\pi}{128}$ respectively.</p> <p>So, the questions are: Is it a known fact? If so, where can I read more on it? If not, may anybody help me to prove/disprove it? Does it mean anything?</p> http://mathoverflow.net/questions/119241/harmonacci-recurrence-and-identities-for-pi/119251#119251 Comment by Victor P Victor P 2013-01-20T06:24:56Z 2013-01-20T06:24:56Z Thanks. Much cleaner and clearer than my attempt. http://mathoverflow.net/questions/119241/harmonacci-recurrence-and-identities-for-pi Comment by Victor P Victor P 2013-01-20T06:24:12Z 2013-01-20T06:24:12Z Aaron - Cute indeed.