A019670 -id:A019670 - cp's OEIS frontend

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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

Offset: 0

Michal Paulovic, Dec 08 2022

The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))
  A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)
  A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...
where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/3))
  h = 1 - 1/2 = 0.5 (A020761)

			0.6141848493043784...

Index entries for transcendental numbers

Cf. A000796, A019670, A020761, A093731, A104954, A120011.

Maple
```
evalf(Pi/3-sqrt(3)/4);
```

RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

PARI
```
Pi/3 - sqrt(3)/4
```

Equals A019670 - A120011. - Omar E. Pol, Dec 08 2022

Equals A093731 / 2. - Michal Paulovic, Mar 08 2024

A371858 Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^7) dx.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.0343760552667964829453064065124887483642567...

Index entries for transcendental numbers.

Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^k) dx: A019669 (k=2), A248897 (k=3), A093954 (k=4), A352324 (k=5), A019670 (k=6), this sequence (k=7), A352125 (k=8).

Cf. A019674, A047336, A121598.

RealDigits[(1/7) Pi Csc[Pi/7], 10, 102][[1]]

Equals (1/7) * Pi * csc(Pi/7).
Equals A019674 * A121598.
Equals Product_{k>=2} (1 + (-1)^k/A047336(k)). - Amiram Eldar, Nov 22 2024

A070988 Continued fraction for Pi/3.

Original entry on oeis.org

1, 21, 5, 3, 97, 1, 1, 5, 13, 1, 4, 7, 1, 3, 8, 255, 5, 5, 9, 4, 1, 1, 1, 1, 6, 2, 18, 33, 4, 4, 1, 1, 2, 3, 1, 1, 5, 1, 1, 2, 1, 2, 3, 2, 1, 1, 3, 3, 1, 1, 1, 9, 1, 1, 3, 1, 7, 2, 1, 26, 1, 2, 1, 1, 1, 1, 5, 1, 1, 4, 12, 13, 1, 2, 1, 4, 1, 485, 15, 3, 7, 1, 1, 3, 4, 1, 1, 1, 3, 1, 7, 1, 1, 2, 1, 2, 9

Offset: 0

Benoit Cloitre, May 18 2002

Cf. A019670 (decimal expansion).

contfrac(Pi/3) \\ Michel Marcus, Nov 26 2013

Offset changed by Andrew Howroyd, Jul 06 2024

A210509 x(n+1) = x(n)*Pi/3 with x(0) = Pi, and a(n) = floor(x(n)).

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 41, 43, 45, 47, 49, 52, 54, 57, 60, 62, 65, 69, 72, 75, 79, 83, 86

Offset: 0

Jon Perry, Jan 25 2013

nonn

24 is the first number greater than 3 not in the sequence.

Later output is more separated, for example: 40888572599, 42818413097, 44839337342, 46955644261.

Cf. A000796 (Pi), A019670 (Pi/3).

pi=Math.PI;
x=pi;
for (i=0;i<600;i++) {x*=pi;x/=3;document.write(Math.floor(x)+", ");}

x(n+1) = x(n)*Pi/3 with x(0) = Pi, and a(n) = floor(x(n)).

A210962 Decimal expansion of 4*(2 - Pi/3).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 03 2012

Volume between a sphere of radius 1 and the circumscribed cube.

			3.8112097952136090153831...

Index entries for transcendental numbers

Cf. A019670, A019699, A153799.

RealDigits[4 (2 - Pi/3), 10, 87][[1]] (* Bruno Berselli, Aug 03 2012 *)

8 - 4/3*Pi \\ Charles R Greathouse IV, Oct 01 2022

4*(2 - Pi/3) = 8 - 4*Pi/3 = 8 - A019699.

A379531 Decimal expansion of (3sqrt(6) - 7)Pi/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 24 2024

Lower bound to the volume of the Meisser's tetrahedral analog of the Reuleaux triangle.

			0.3649161225950003501847169303738650723435020735...

G. D. Chakerian and H. Groemer, Convex bodies of constant width, Convexity and Its Applications, ed. P. M. Gruber and J. M. Wills, Birkhäuser, 1983, pp. 49-96; MR85f:52001.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.10, p. 514.

G. D. Chakerian, Sets of constant width, Pacific J. Math. 19 (1966) 13-21; MR 34 #4986. See p. 14.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 230.

Cf. A000796, A010464, A010507, A019670.

RealDigits[(3Sqrt[6]-7)Pi/3,10,100][[1]]

cp's OEIS Frontend

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A371858 Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^7) dx.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A070988 Continued fraction for Pi/3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

A210509 x(n+1) = x(n)*Pi/3 with x(0) = Pi, and a(n) = floor(x(n)).

Views

Author

Keywords

Comments

Crossrefs

Programs

JavaScript

Formula

A210962 Decimal expansion of 4*(2 - Pi/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A379531 Decimal expansion of (3*sqrt(6) - 7)*Pi/3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A379531 Decimal expansion of (3sqrt(6) - 7)Pi/3.