cp's OEIS Frontend

A350843 The least number of terms needed in the Taylor series approximation of arctan(1/239) such that Machin's formula with n terms in the Taylor series approximation of arctan(1/5) achieves the most correct digits of Pi.

%I A350843 #23 Dec 23 2023 11:11:34
%S A350843 1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,6,6,7,7,7,8,8,8,8,9,9,9,10,
%T A350843 10,10,11,11,11,11,12,12,12,13,13,13,14,14,14,14,15,15,16,16,16,16,17,
%U A350843 17,17,17,18,18,18,19,19,19,19,20,20,20
%N A350843 The least number of terms needed in the Taylor series approximation of arctan(1/239) such that Machin's formula with n terms in the Taylor series approximation of arctan(1/5) achieves the most correct digits of Pi.
%C A350843 Machin's formula states that Pi/4 = 4*arctan(1/5) - arctan(1/239). An approximation of Pi can be found by computing this using a Taylor series approximation of arctan. If n terms are used in the approximation of arctan(1/5), then a(n) is the least number of terms that can be used in the approximation of arctan(1/239) to get the largest possible number of correct digits of Pi.
%H A350843 Matthew Scroggs, <a href="https://github.com/mscroggs/machins-formula/blob/main/A350843.py">Python code</a>
%H A350843 Wikipedia, <a href="https://en.wikipedia.org/wiki/John_Machin">John Machin</a>
%e A350843 When using 5 terms in the Taylor series expansion of arctan(1/5) and 2 terms in the expansion of arctan(1/239), Machin's formula gives 3.141592682405... which is correct to 7 decimal places. If more than 2 terms are used in the second expansion, no more correct digits are obtained. If fewer than 2 terms are used, fewer correct digits will be obtained. Therefore a(5) = 2.
%Y A350843 Cf. A000796, A096954, A096955.
%Y A350843 A350799(n) is the number of decimal places that will be correct when n terms are used for arctan(1/5) and a(n) terms are used for arctan(1/239).
%K A350843 nonn,base,more
%O A350843 1,5
%A A350843 _Matthew Scroggs_, Jan 18 2022