A096955 -id:A096955 - cp's OEIS frontend

A096954 Numerators of rational approximation to Pi/4 from Machins's formula.

951, 1339849258, 9569810428334921, 19132121777295048135244, 81963468350564671450762204559, 1287504688596138051498743351405666674, 23901655485793371607250742363386659018053931

Offset: 0

Wolfdieter Lang, Jul 23 2004

Machin's formula: Pi/4 = 4*arctan(1/5) - arctan(1/239).

Denominators are given in A096955.

			a(2)/A096955(2) = 9569810428334921/12184551018734375 = .78540...

W. Walter, Analysis I (in German), Springer, 3. Auflage, 1992; p. 216.

Wolfdieter Lang, More comments.
Eric Weisstein's World of Mathematics, Machin's Formula.

Cf. A003881, A096955.

a(n)=numerator(M(n)), with M(n)=4*arctan(1/5, n) - arctan(1/239, n) with arctan(x, n):=sum((((-1)^k)*x^(2k+1))/(2*k+1), k=0..n).

A278364 A sequence showing denominators in ratios tending to the constant Pi/4 = 0.785398163397448... .

Original entry on oeis.org

5, 375, 46875, 1640625, 123046875, 33837890625, 10997314453125, 1374664306640625, 116846466064453125, 55502071380615234375

Offset: 1

Sanjar Abrarov, Dec 04 2016

The ratios c(n)/d(n) rapidly tend to the constant Pi/4 = 0.785398163397448... with increasing index n: abs(Pi/4 - c(1)/d(1)) > abs(Pi/4 - c(2)/d(2)) > abs(Pi/4 - c(3)/d(3)) > abs(Pi/4 - c(4)/d(4)) ..., where c(n) = A278924(n) and d(n) = A278364(n) are even and odd positive integers, respectively. All denominators d(n) are divisible by 5.

			------------------------------------------------
n    c(n)                   d(n)
------------------------------------------------
1    4                      5
2    296                    375
3    36772                  46875
4    1288688                1640625
5    96641548               123046875
6    26576092808            33837890625
7    8637277012172          10997314453125
8    1079658805128928       1374664306640625
9    91770997994914276      116846466064453125
10   43591225139846360008   55502071380615234375
------------------------------------------------
At n = 6 the ratio c(6)/d(6) = 26576092808/33837890625 is close to Pi/4. However, at n = 10 the ratio c(10)/d(10) = 43591225139846360008/55502071380615234375 becomes more closer to Pi/4.

Sanjar Abrarov, Table of n, a(n) for n = 1..49
S. M. Abrarov and B. M. Quine, A generalized Viéte's-like formula for pi with rapid convergence, arXiv:1610.07713 [math.GM], (2016).

Cf. A278924, A003881, A096954, A096955.

x := 1; (* argument x *)
M := 1; (* initial value for the integer M *)
n := 1; (* index *)
(* Note that arctan(1) = Pi/4 *)
atan := I*Sum[(1/(2*m - 1))*(1/(1 + 2*(I/x))^(2*m - 1) - 1/(1 - 2*(I/x))^(2*m - 1)), {m, 1, Floor[M/2] + 1}];
sqn := {}; (* initiate the sequence *)
AppendTo[sqn, {"Index n", "Numerators", "Denominators"}];
While[M <= 20, AppendTo[sqn, {n, Numerator[atan], Denominator[atan]}];
{M = M + 2, n++}];
Print[MatrixForm[sqn]]

arctan(1) = I*lim_{M -> inf}Sum_{m = 1..floor(M/2) + 1}(1/(2*m - 1))*(1/(1 + 2*I)^(2*m - 1) - 1/(1 - 2*I)^(2*m - 1))

A278924 A sequence showing numerators in ratios tending to the constant Pi/4 = 0.785398163397448... .

Original entry on oeis.org

4, 296, 36772, 1288688, 96641548, 26576092808, 8637277012172, 1079658805128928, 91770997994914276, 43591225139846360008

Offset: 1

Sanjar Abrarov, Dec 01 2016

The ratios c(n)/d(n) rapidly tend to the constant Pi/4 = 0.785398163397448... with increasing index n: abs(Pi/4 - c(1)/d(1)) > abs(Pi/4 - c(2)/d(2)) > abs(Pi/4 - c(3)/d(3)) > abs(Pi/4 - c(4)/d(4)) ..., where c(n) = A278924(n) and d(n) = A278364(n) are even and odd positive integers, respectively.

			------------------------------------------------
n    c(n)                   d(n)
------------------------------------------------
1    4                      5
2    296                    375
3    36772                  46875
4    1288688                1640625
5    96641548               123046875
6    26576092808            33837890625
7    8637277012172          10997314453125
8    1079658805128928       1374664306640625
9    91770997994914276      116846466064453125
10   43591225139846360008   55502071380615234375
------------------------------------------------
At n = 6 the ratio c(6)/d(6) = 26576092808/33837890625 is close to Pi/4. However, at n = 10 the ratio c(10)/d(10) = 43591225139846360008/55502071380615234375 becomes more closer to Pi/4.

Sanjar Abrarov, Table of n, a(n) for n = 1..49
S. M. Abrarov and B. M. Quine, A generalized Viéte's-like formula for pi with rapid convergence, arXiv:1610.07713 [math.GM], (2016).

Cf. A278364, A003881, A096954, A096955.

x := 1;(* argument x *)
M := 1;(* initial value for the integer M *)
n := 1; (* index *)
(* Note that arctan(1) = Pi/4 *)
atan := I*Sum[(1/(2*m - 1))*(1/(1 + 2*(I/x))^(2*m - 1) - 1/(1 - 2*(I/x))^(2*m - 1)),{m, 1, Floor[M/2] + 1}];
sqn := {};(* initiate the sequence *)
AppendTo[sqn,{"Index n", "Numerators", "Denominators"}];
While[M <= 20, AppendTo[sqn,{n, Numerator[atan], Denominator[atan]}];
{M = M + 2, n++}];
Print[MatrixForm[sqn]]

arctan(1) = I*lim_{M -> inf}Sum_{m = 1..floor(M/2) + 1}(1/(2*m - 1))*(1/(1 + 2*I)^(2*m - 1) - 1/(1 - 2*I)^(2*m - 1))

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

Original entry on oeis.org

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

Matthew Scroggs, Jan 18 2022

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.

			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.

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

Cf. A000796, A096954, A096955.

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.

Original entry on oeis.org

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

Offset: 1

Matthew Scroggs, Jan 18 2022

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.

			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.

Matthew Scroggs, Python code
Wikipedia, John Machin

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).

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A096954 Numerators of rational approximation to Pi/4 from Machins's formula.

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A278364 A sequence showing denominators in ratios tending to the constant Pi/4 = 0.785398163397448... .

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A278924 A sequence showing numerators in ratios tending to the constant Pi/4 = 0.785398163397448... .

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

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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.

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