A068028 -id:A068028 - cp's OEIS frontend

A255873 The first nonzero digit of n/7.

1, 2, 4, 5, 7, 8, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Kit Scriven, Mar 08 2015

			The leading (most significant) digit of 22/7 in A068028 is 3, so a(22)=3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A020806 (1/7), A068028 (22/7), A216606 (360/7).

A255873 := proc(n)
    local nshf ;
    nshf := n/7 ;
    while nshf < 1 do
        nshf := 10*nshf;
    end do;
    while nshf >= 10 do
        nshf := nshf/10;
    end do;
    floor(nshf) ;
end proc: # R. J. Mathar, May 28 2016

f[n_] := RealDigits[n/7, 10, 9][[1, 1]]; Array[f, 105] (* Robert G. Wilson v, Mar 08 2015 *)

a(n) = {my(x = n/7.0); if (x < 1, x *= 10); while (x >= 10, x /= 10); floor(x);} \\ Michel Marcus, Mar 12 2015

A374322 Decimal expansion of sqrt(2)*9801/4412.

Original entry on oeis.org

3, 1, 4, 1, 5, 9, 2, 7, 3, 0, 0, 1, 3, 3, 0, 5, 6, 6, 0, 3, 1, 3, 9, 9, 6, 1, 8, 9, 0, 2, 5, 2, 1, 5, 5, 1, 8, 5, 9, 9, 5, 8, 1, 6, 0, 7, 1, 1, 0, 0, 3, 3, 5, 5, 9, 6, 5, 6, 5, 3, 6, 2, 9, 0, 1, 2, 8, 5, 5, 1, 4, 5, 5, 4, 4, 1, 3, 2, 1, 6, 4, 2, 7, 4, 0, 8, 5, 4, 0, 8

Offset: 1

Paolo Xausa, Jul 04 2024

Approximates Pi, correct to 7 digits.

			3.1415927300133056603139961890252155185995816071100335596565...

Paolo Xausa, Table of n, a(n) for n = 1..10000
David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the American Mathematical Society, May 2005, Vol. 52, No. 5, p. 511.
Jonathan M. Borwein, The Experimental Mathematician: the Pleasure of Discovery and the Role of Proof, International Journal of Computers for Mathematical Learning, Vol. 10 (2005), p. 97.
Index entries for sequences related to the number Pi

Cf. A000796, A002193, A068028, A068079, A068089, A253214.

First[RealDigits[Sqrt[2]*9801/4412, 10, 100]]

from math import isqrt
def A374322(n): return isqrt(10**(n-1<<1)*96059601//9732872)%10 # Chai Wah Wu, Jul 04 2024

A380109 Decimal expansion of 223/71.

Original entry on oeis.org

3, 1, 4, 0, 8, 4, 5, 0, 7, 0, 4, 2, 2, 5, 3, 5, 2, 1, 1, 2, 6, 7, 6, 0, 5, 6, 3, 3, 8, 0, 2, 8, 1, 6, 9, 0, 1, 4, 0, 8, 4, 5, 0, 7, 0, 4, 2, 2, 5, 3, 5, 2, 1, 1, 2, 6, 7, 6, 0, 5, 6, 3, 3, 8, 0, 2, 8, 1, 6, 9, 0, 1, 4, 0, 8, 4, 5, 0, 7, 0, 4, 2, 2, 5, 3, 5, 2, 1, 1, 2, 6, 7, 6, 0, 5, 6, 3, 3, 8, 0

Offset: 1

Stefano Spezia, Jan 12 2025

This constant is a lesser limit to the value of Pi calculated by Archimedes considering 96-gons.

Apart from the first digit the same as A021075. - R. J. Mathar, Jan 17 2025

			3.1408450704225352112676056338028169014084507042...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi and §13.3 Solving Triangles, pp. 90, 479.

Cf. A000796, A021075, A068028 (greater limit).

Mathematica
```
RealDigits[3+10/71,10,100][[1]]
```

Equals 3 + 10/71.

A021319 Decimal expansion of 1/315.

Original entry on oeis.org

0, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3, 1, 7, 4, 6, 0, 3

Offset: 0

N. J. A. Sloane

If the initial 0 is ignored, a(n) is periodic with period 6: [0, 3, 1, 7, 4, 6]. - Wesley Ivan Hurt, Oct 10 2014

			1/315 = 0.0031746031746031746031746031...

Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 1).

Cf. A020806, A068028.

Digits:=100: evalf(1/315); # Wesley Ivan Hurt, Oct 10 2014

RealDigits[1/315, 10, 100, -1][[1]] (* Wesley Ivan Hurt, Oct 10 2014 *)
Join[{0},LinearRecurrence[{1, 0, -1, 1},{0, 3, 1, 7},98]] (* Ray Chandler, Aug 26 2015 *)

a(n) = a(n-1)-a(n-3)+a(n-4) for n>0, with a(0)=0; a(n) = A068028(n+1)-1 for n>0; a(n+1) = A020806(n)-1. - Wesley Ivan Hurt, Oct 10 2014

A339264 Decimal expansion of (63/25) * (17+15sqrt(5)) / (7+15sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.

Original entry on oeis.org

3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 8, 0, 5, 6, 8, 8, 2, 0, 1, 8, 9, 8, 3, 9, 0, 0, 0, 6, 3, 0, 1, 5, 0, 7, 8, 2, 2, 4, 8, 7, 5, 0, 3, 4, 7, 5, 7, 7, 4, 3, 0, 9, 2, 2, 2, 8, 3, 8, 6, 6, 0, 9, 2, 8, 2, 2, 0, 4, 2, 4, 6, 3, 7, 4, 4, 5, 2, 5, 1, 1, 6, 3, 5, 4, 8, 9, 2, 9, 9, 6

Offset: 1

Bernard Schott, Nov 29 2020

This formula that derives from Ramanujan modular equations is correct to 9 places exactly (see Ramanujan link, page 43).
Pi = 3.1415926535... and this approximation = 3.1415926538...
A quadratic number with minimal polynomial 168125x^2 - 792225x + 829521 and denominator 6725. - Charles R Greathouse IV, Oct 02 2022

			3.141592653805688201898390006301507822487503475774...

Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved Jun 05 2013, (4.17) page 57.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.

S. Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, p. 43.
Index entries for algebraic numbers, degree 2

Other approximations to Pi: A068028, A068079, A068089, A328927.

evalf((63/25)*(17+15*sqrt(5))/(7+15*sqrt(5)),100);

RealDigits[(63/25)*(17 + 15*Sqrt[5])/(7 + 15*Sqrt[5]), 10, 100][[1]] (* Amiram Eldar, Nov 29 2020 *)

(63/13450) * (503+75*sqrt(5)) \\ Michel Marcus, Nov 29 2020

Equals (63/13450) * (503+75*sqrt(5)).

Equals the root of 829521 - 792225*x + 168125*x^2 which is > 3. - Peter Luschny, Nov 29 2020

A342977 Decimal expansion of (Pi - 2) / 4.

Original entry on oeis.org

2, 8, 5, 3, 9, 8, 1, 6, 3, 3, 9, 7, 4, 4, 8, 3, 0, 9, 6, 1, 5, 6, 6, 0, 8, 4, 5, 8, 1, 9, 8, 7, 5, 7, 2, 1, 0, 4, 9, 2, 9, 2, 3, 4, 9, 8, 4, 3, 7, 7, 6, 4, 5, 5, 2, 4, 3, 7, 3, 6, 1, 4, 8, 0, 7, 6, 9, 5, 4, 1, 0, 1, 5, 7, 1, 5, 5, 2, 2, 4, 9, 6, 5, 7, 0, 0, 8, 7, 0, 6, 3, 3, 5, 5, 2, 9, 2

Offset: 0

Michal Paulovic, Apr 01 2021

The constant represents the area of a circular segment bounded by an arc of Pi/2 radians (the right angle) of a unit circle and by a chord of the length of sqrt(2). Four such segments result when a square with the side length of sqrt(2) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (Pi/2 - sin(Pi/2))
  A = (1 / 2) * (Pi/2 - 1)
  A = (Pi - 2) / 4 = 0.28539816...
  where Pi = 3.14159265... (A000796) is the area bounded by the unit circle, and 2 is the area of the inscribed square.
Apart from the first digit this is the same as Pi/4 = 0.78539816... (A003881), the area of a circular sector bounded by the arc of Pi/2 = 1.57079632... (A019669) radians of the unit circle and by two radii of unit length, and 1/2 = 0.5 (A020761) is one-quarter of the area of the inscribed square.
The constant is close to 2/7 = 0.28571428... (2 * A020806) and Pi/11 = 0.28559933... (A019678). The equation (x - 2)/4 = x/11 has a solution x = 22/7 = 3.14285714... (A068028), which is an approximation of Pi.
The best rational approximation of the constant using small positive integers (less than 1000) is 129/452 = 0.28539823..., the next best approximation is 4771/16717 = 0.28539809...
The reciprocal of the constant is:
  1/A = 4 / (Pi - 2) = 3.50387678... (A309091).
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/4))
  h = 1 - sqrt(2) / 2
  h = 1 - 1 / sqrt(2) = 0.29289321... (A268682).

			0.2853981633974483...

Index entries for transcendental numbers

Cf. A000796, A019669, A019678, A020761, A020806, A068028, A268682, A309091. Essentially the same as A003881.

RealDigits[Pi/4 - 1/2, 10, 100][[1]] (* Amiram Eldar, Jun 08 2021 *)

PARI
```
(Pi - 2) / 4
```

Equals Integral_{x=-sqrt(2)/2..sqrt(2)/2} Integral_{y=sqrt(2)/2..sqrt(1-x^2)} dy dx.
Equals Sum_{k>=1} (-1)^(k + 1)/(4*k^2 - 1). - Amiram Eldar, Jun 08 2021
Continued fraction: 1/(3 + 3/(4 + 15/(4 + 35/(4 + ... + (4*n^2 - 1)/(4 + ...). - Peter Bala, Feb 22 2024

A380152 Decimal expansion of 864/275.

Original entry on oeis.org

3, 1, 4, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1

Offset: 1

Stefano Spezia, Jan 13 2025

This approximation of Pi was calculated by Fibonacci studying the 96-gon and published in 1220 in Practica geometriae (see Gullberg).

			3.1418181818181818181818181818181818181818181818...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, p. 91.

Cf. A000796, A068028, A380109.

Mathematica
```
RealDigits[864/275,10,100][[1]]
```

cp's OEIS Frontend

A255873 The first nonzero digit of n/7.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A374322 Decimal expansion of sqrt(2)*9801/4412.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A380109 Decimal expansion of 223/71.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

A021319 Decimal expansion of 1/315.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A339264 Decimal expansion of (63/25) * (17+15*sqrt(5)) / (7+15*sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A342977 Decimal expansion of (Pi - 2) / 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A380152 Decimal expansion of 864/275.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A339264 Decimal expansion of (63/25) * (17+15sqrt(5)) / (7+15sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.