A165922 -id:A165922 - cp's OEIS frontend

A049541 Decimal expansion of 1/Pi.

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

Offset: 0

N. J. A. Sloane, Dec 11 1999

The ratio of the volume of a regular octahedron to the volume of the circumscribed sphere (which has circumradius a*sqrt(2)/2 = a*A010503, where a is the octahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A165952, A165953 and A165954. - Rick L. Shepherd, Oct 01 2009

Corresponds to a gauge point marked "M" on slide rule calculating devices in the 20th century. The Pickworth reference notes its use in calculating the area of the curved surface of a cylinder. - Peter Munn, Aug 14 2020

			0.3183098861837906715377675267450287240689192914809128974953...

J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Baasel, p. 245. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London, 1945, p. 53, Gauge Points.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
J. Bohr, Ramanujan's Method of Approximating Pi.
J. Borwein, Ramanujan's Sum.
Heng Huat Chan, Shaun Cooper, and Wen-Chin Liaw, The Rogers-Ramanujan continued fraction and a quintic iteration for 1/Pi, Proc. Amer. Math. Soc. 135 (2007), 3417-3424.
D. V. Chudnovsky and G. V. Chudnovsky, The computation of classical constants, Proc. Nati. Acad. Sci. USA, Vol. 86, pp. 8178-8182, November 1989.
J. Guillera, A New Method to Obtain Series for 1/Pi and 1/Pi^2, Experimental Mathematics, Volume 15, Issue 1, 2006.
R. Matsumoto, Ramanujan Type Series. [Broken link]
A. S. Nimbran, Deriving Forsyth-Glaisher type series for 1/π and Catalan's constant by an elementary method, The Mathematics Student, Indian Math. Soc., Vol. 84, Nos. 1-2, Jan.-June (2015), 69-86. [Broken link]
Eric W. Weisstein, Octahedron.
Index entries for transcendental numbers.

Cf. A000796, A165922, A165952, A165953, A165954, A063723, A010503. - Rick L. Shepherd, Oct 01 2009

Cf. A088538 (4/Pi).

MATLAB
```
1/pi % Altug Alkan, Apr 10 2016
```

R:= RealField(100); 1/Pi(R); // G. C. Greubel, Aug 21 2018

Digits:=100: evalf(1/Pi); # Wesley Ivan Hurt, Aug 29 2016

RealDigits[N[1/Pi, 10, 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

1/Pi \\ Charles R Greathouse IV, Jun 16 2011

Equals (1/(12-16*A002162))*Sum_{n>=0} A002894(n)*H(n)/(A001025(n) * A016754(n-1)), where H(n) denotes the n-th harmonic number. - John M. Campbell, Aug 28 2016
1/Pi = Sum_{m>=0} binomial(2*m, m)^3 * (42*m+5)/(2^(12*m+4)), Ramanujan, from the J.-P. Delahaye reference. - Wolfdieter Lang, Sep 18 2018; corrected by Bernard Schott, Mar 26 2020
1/Pi = 12*Sum_{n >= 0} (-1)^n*((6*n)!/(n!^3*(3*n)!))*(13591409 + 545140134*n)/640320^(3*n + 3/2) [Chudnovsky]. - Sanjar Abrarov, Mar 31 2020
1/Pi = (sqrt(8)/9801) * Sum_{n >= 0} ((4*n)!/((n!)^4)) * (26390*n + 1103)/(396^(4*n)) [Ramanujan, 1914]. - Bernard Schott, Mar 26 2020
Equal Sum_{k>=2} tan(Pi/2^k)/2^k. - Amiram Eldar, Aug 05 2020
Floor((3/8)*Sum_{n>=1} sigma[3](n)*n/exp(Pi*n/(10^((1/5)*k+(1/5))))) mod 10, will give the k-th digit of 1/Pi. - Simon Plouffe, Dec 19 2023

A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Original entry on oeis.org

4, 8, 6, 20, 12

Offset: 1

Henry Bottomley, Aug 14 2001

The preferred order for these five numbers is 4, 6, 8, 12, 20 (tetrahedron, octahedron, cube, icosahedron, dodecahedron), as in A053016. - N. J. A. Sloane, Nov 05 2020
Also number of faces of Platonic solids ordered by increasing ratios of volumes to their respective circumscribed spheres. See cross-references for actual ratios. - Rick L. Shepherd, Oct 04 2009
Also the expected lengths of nontrivial random walks along the edges of a Platonic solid from one vertex back to itself. - Jens Voß, Jan 02 2014

			a(2) = 8 since a cube has eight vertices.

Cf. A053012, A053016, A060296, A060852.
Cf. A165922 (tetrahedron), A049541 (octahedron), A165952 (cube), A165954 (icosahedron), A165953 (dodecahedron). - Rick L. Shepherd, Oct 04 2009
Cf. A234974. - Jens Voß, Jan 02 2014

a(n) = A063722(n) - A053016(n) + 2.

A165952 Decimal expansion of 2sqrt(3)/(3Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a cube to the volume of the circumscribed sphere (which has circumradius a*sqrt(3)/2 = a*A010527, where a is the cube's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165953, and A165954. A063723 shows the order of these by size.

			0.3675525969478613663408843322086462942649243202444271018662440135273535356...

Eric Weisstein's World of Mathematics, Cube.
Index entries for transcendental numbers

Cf. A000796, A002194, A165922, A049541, A165953, A165954, A063723, A020760, A020832.

RealDigits[(2*Sqrt[3])/(3Pi),10,120][[1]] (* Harvey P. Dale, Oct 08 2012 *)

PARI
```
2*sqrt(3)/(3*Pi)
```

2*sqrt(3)/(3*Pi) = 2*A002194/(3*A000796) = 3*A165922 = (2*sqrt(3)/3)*A049541 = 10*A020832*A049541 = 2*A020760*A049541.

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.

			0.6649088942053266431144284467086337161648765805556919381057592605722964718...

Eric Weisstein's World of Mathematics, Dodecahedron.
Index entries for transcendental numbers.

Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.

RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi),10,120][[1]] (* Harvey P. Dale, Feb 16 2018 *)

PARI
```
(5*sqrt(3)+sqrt(15))/(6*Pi)
```

Equals (5*A002194 + A010472)/(6*A000796).
Equals (5*A002194 + A010472)*A049541/6.
Equals (10*A010527 + A010472)*A049541/6.
Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).
Equals (5 + A002163)*A049541*A020760/2.

A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 04 2009

The ratio of the volume of a regular icosahedron to the volume of the circumscribed sphere (with circumradius a*sqrt(10 + 2*sqrt(5))/4 = a*A019881, where a is the icosahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165953. A063723 shows the order of these by size.

			0.6054613829125255833862652051280444903008454088014288933200935000838295683...

Eric Weisstein's World of Mathematics, Icosahedron.
Index entries for transcendental numbers.

Cf. A000796, A002163, A165922, A049541, A165952, A165953, A063723, A019881, A019694, A060294.

RealDigits[Sqrt[10+2Sqrt[5]]/(2Pi),10,120][[1]] (* Harvey P. Dale, Aug 27 2013 *)

PARI
```
sqrt(10+2*sqrt(5))/(2*Pi)
```

sqrt(10 + 2*sqrt(5))/(2*Pi) = sqrt(10 + 2*A002163)/(2*A000796) = 2*sin(2*Pi/5)/Pi = 2*sin(A019694)/A000796 = 2*sin(72 deg)/Pi = 2*A019881/A000796 = 2*A019881*A049541 = (2/Pi)*sin(72 deg) = A060294*A019881.

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A049541 Decimal expansion of 1/Pi.

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A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

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A165952 Decimal expansion of 2*sqrt(3)/(3*Pi).

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A165953 Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).

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A165954 Decimal expansion of sqrt(10 + 2*sqrt(5))/(2*Pi).

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A165952 Decimal expansion of 2sqrt(3)/(3Pi).

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

A165954 Decimal expansion of sqrt(10 + 2sqrt(5))/(2Pi).