Wallis' product exploration

02092020, 01:08 PM
Post: #9




RE: Wallis' product exploration
What I find mind blowing about it is that it's a combinatoric function, not of just triangles, but how many different ways triangles can tile a nearlycircular polygon.
I tried yesterday to find a printed/published formula for \( \pi \) in terms of catalan numbers, but could not. combining terms with Albert's observation above, we get \(\pi = \lim\limits_{n\to\infty} \frac {2^{4n+2} } { (2n+1) (n+1)^2 C_n^2} \) interesting scaling polynomial in the bottom \( 2 n^3 + 5 n^2 + 4 n + 1 \) Out of curiosity, I think we can replace the \( (n+1)^2\) with a Triangluar number \( T_n\) and further drop a factor of 4 from the numerator: \( \pi = \lim\limits_{n\to\infty} \frac {n^2 2^{4n} } { (2n+1) T_n^2 C_n^2} \) in effort to reduce some polynomial terms, but I can't tell if that's a step forward or a step back.. Ideally, one could represent \( \pi \) as an integer divided by a hand full of (discrete) combinatoric functions. 17bii  32s  32sii  41c  41cv  41cx  42s  48g  48g+  48gx  50g  30b 

« Next Oldest  Next Newest »

User(s) browsing this thread: 1 Guest(s)