A350799 The number of decimal places of Pi that are computed correctly when using Machin's formula with n terms of the Taylor series.

1, 2, 3, 5, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 21, 22, 24, 25, 27, 29, 30, 30, 32, 34, 36, 37, 38, 40, 40, 43, 42, 45, 47, 47, 49, 51, 53, 54, 55, 57, 58, 59, 60, 62, 64, 65, 67, 68, 69, 71, 72, 74, 75, 75, 77, 79, 80, 82

Offset: 1

Author

Matthew Scroggs, Jan 18 2022

Keywords

• nonn
• base

Comments

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. a(n) is the number of decimal places that are correct when n terms are included in the Taylor series approximation.

Examples

			For n = 3, Machin's formula with three terms in the Taylor series gives 3.14162102932503442504 as an approximation of Pi. The first 3 decimal places (141) are correct, so a(3) = 3.
		

Links

• Matthew Scroggs, Table of n, a(n) for n = 1..200
• Matthew Scroggs, Python code
• Wikipedia, John Machin

Crossrefs

Cf. A000796, A096954, A096955.