Wallis' product exploration
02-09-2020, 01:08 PM
Post: #9
 Allen Member Posts: 227 Joined: Aug 2014
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 nearly-circular 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

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 Messages In This Thread Wallis' product exploration - pinkman - 02-07-2020, 10:55 PM RE: Wallis' product exploration - Albert Chan - 02-08-2020, 02:13 AM RE: Wallis' product exploration - pinkman - 02-08-2020, 02:42 PM RE: Wallis' product exploration - Allen - 02-08-2020, 07:13 PM RE: Wallis' product exploration - Albert Chan - 02-08-2020, 07:53 PM RE: Wallis' product exploration - Allen - 02-08-2020, 08:58 PM RE: Wallis' product exploration - pinkman - 02-09-2020, 05:44 AM RE: Wallis' product exploration - Allen - 02-08-2020, 08:13 PM RE: Wallis' product exploration - Allen - 02-09-2020 01:08 PM RE: Wallis' product exploration - Allen - 02-09-2020, 01:57 PM RE: Wallis' product exploration - Allen - 02-09-2020, 03:08 PM RE: Wallis' product exploration - pinkman - 02-09-2020, 02:14 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-09-2020, 10:58 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 12:53 AM RE: Wallis' product exploration - EdS2 - 02-10-2020, 10:35 AM RE: Wallis' product exploration - pinkman - 02-11-2020, 10:02 AM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 04:11 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-11-2020, 10:48 PM RE: Wallis' product exploration - pinkman - 02-12-2020, 10:01 PM RE: Wallis' product exploration - Gerson W. Barbosa - 02-13-2020, 12:17 AM

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