"Harmonacci" recurrence and identities for $\pi$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:40:31Z http://mathoverflow.net/feeds/question/119241 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 Answer by Felix Janda for "Harmonacci" recurrence and identities for $\pi$ Felix Janda 2013-01-18T10:42:48Z 2013-01-18T11:57:45Z <p>The sequence $a_n$ is closely related to the Wallis product <code>$$a'_n = \prod_{i = 1}^n \left(\frac{2i}{2i - 1} \frac{2i}{2i + 1}\right),$$</code> which converges to $\pi/2$ as $n$ goes to infinity. Namely, we have <code>$$a'_n = a_{n + 1} \cdot \frac{2n}{2n + 1}$$</code>. This could be proven by induction or maybe more easily by defining $b_n = a_n a_{n - 1}$ and noticing that the recursion for $a_n$ implies the (very simple) recursion $$b_{n + 1} = 1 + b_n$$ for $b_n$ and expressing $a_n$ in terms of the $b_n$.</p> <p>For more general values of $a_0$ one gets similar formulas for $a_n$ as (up to a factor converging to 1) a Wallis product or inverse of a Wallis product where a few of the lower terms in the product are missing.</p>