A350799 -id:A350799

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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.

+10
1

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, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20

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OFFSET

1,5

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. 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.

LINKS

Table of n, a(n) for n=1..69.

Matthew Scroggs, Python code

Wikipedia, John Machin

EXAMPLE

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.

CROSSREFS

Cf. A000796, A096954, A096955.

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).

KEYWORD

nonn,base,more

AUTHOR

Matthew Scroggs, Jan 18 2022

STATUS

approved

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