A165952 -id:A165952 - 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

A020760 Decimal expansion of 1/sqrt(3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

If the sides of a triangle form an arithmetic progression in the ratio 1:1+d:1+2d then when d=1/sqrt(3) it uniquely maximizes the area of the triangle. This triangle has approximate internal angles 25.588 degs, 42.941 degs, 111.471 degs. - Frank M Jackson, Jun 15 2011
When a cylinder is completely enclosed by a sphere, it occupies a fraction f of the sphere volume. The value of f has a trivial lower bound of 0, and an upper bound which is this constant. It is achieved iff the cylinder diameter is sqrt(2) times its height, and the sphere is circumscribed to it. A similar constant can be associated with any n-dimensional geometric shape. For 3D cuboids it is A165952. - Stanislav Sykora, Mar 07 2016
The ratio between the thickness and diameter of a dynamically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on the dynamic of rigid body (Yong and Mahadevan, 2011). See A020765 for a simplified geometrical solution. - Amiram Eldar, Sep 01 2020
The coefficient of variation (relative standard deviation) of natural numbers: Limit_{n->oo} sqrt((n-1)/(3*n+3)) = 1/sqrt(3). - Michal Paulovic, Mar 21 2023

			0.577350269189625764509148780501957455647601751270126876018602326....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.17, pp. 495, 531.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ee Hou Yong and L. Mahadevan, Probability, geometry, and dynamics in the toss of a thick coin, American Journal of Physics, Vol. 79, No. 12 (2011), pp. 1195-1201, preprint, arXiv:1008.4559 [physics.class-ph], 2010-2011.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (sqrt(3)), A010701 (1/3).

Cf. A002193, A047235, A165952, A195696, A040001.

evalf(1/sqrt(3)); # Michal Paulovic, Mar 21 2023

RealDigits[N[1/Sqrt[3],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

\\ Works in v2.15.0; n = 100 decimal places
my(n=100); digits(floor(10^n/quadgen(12))) \\ Michal Paulovic, Mar 21 2023

Equals 1/A002194. - Michel Marcus, Oct 12 2014
Equals cosine of the magic angle: cos(A195696). - Stanislav Sykora, Mar 07 2016
Equals square root of A010701. - Michel Marcus, Mar 07 2016
Equals 1 + Sum_{k>=0} -(4*k+1)^(-1/2) + (4*k+3)^(-1/2) + (4*k+5)^(-1/2) - (4*k+7)^(-1/2). - Gerry Martens, Nov 22 2022
Equals (1/2)*(2 - zeta(1/2,1/4) + zeta(1/2,3/4) + zeta(1/2,5/4) - zeta(1/2,7/4)). - Gerry Martens, Nov 22 2022
Has periodic continued fraction expansion [0, 1; 1, 2] (A040001). - Michal Paulovic, Mar 21 2023
Equals Product_{k>=1} (1 + (-1)^k/A047235(k)). - Amiram Eldar, Nov 22 2024
Equals tan(Pi/6) = (1/2)/A010527. - R. J. Mathar, Aug 31 2025

A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane

Number of elliptic points of order 3 for Gamma_0(n).
Equivalently, number of fixed points of Gamma_0(n) of type rho.
Values are 0 or a power of 2.
Shadow transform of central polygonal numbers A002061. - Michel Marcus, Jun 06 2013
Empirical: a(n) == A001615(n) (mod 3) for all natural numbers n. - John M. Campbell, Apr 01 2018
From Jianing Song, Jul 03 2018: (Start)
The comment above is true. Since both a(n) and A001615(n) are multiplicative we just have to verify that for prime powers. Note that A001615(p^e) = (p+1)*p^(e-1). For p == 1 (mod 3), p+1 == 2 (mod 3) so (p+1)*p^(e-1) == 2 (mod 3); for p == 2 (mod 3), p+1 is a multiple of 3 so (p+1)*p^(e-1) == 0 (mod 3). For p = 3, if e = 1 then p+1 == 1 (mod 3); if e > 1 then (p+1)*p^(e-1) == 0 (mod 3).
Equivalently, number of solutions to x^2 + x + 1 == 0 (mod n). (End)

			G.f. = x + x^3 + 2*x^7 + 2*x^13 + 2*x^19 + 2*x^21 + 2*x^31 + 2*x^37 + 2*x^39 + ...

Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).

Christian G. Bower, Table of n, a(n) for n = 1..2000
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 61-68.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. A65 (2009), 156-163. [See Table 4.]
N. J. A. Sloane, Transforms.

Cf. A000089, A000091, A001616, A014683.
Cf. A027748, A079978, A038502, A007949, A165952.
Cf. A341422 (without zeros).

a000086 n = if n `mod` 9 == 0 then 0
  else product $ map ((* 2) . a079978 . (+ 2)) $ a027748_row $ a038502 n
-- Reinhard Zumkeller, Jun 23 2013

with(numtheory); A000086 := proc (n) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end: # Gene Ward Smith, May 22 2006

Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
a[ n_] := If[ n < 1, 0, Length[ Select[ (#^2 - # + 1)/n & /@ Range[n], IntegerQ]]]; (* Michael Somos, Aug 14 2015 *)
a[n_] := a[n] = Product[{p, e} = pe; Which[p==1 || p==3 && e==1, 1, p==3 && e>1, 0, Mod[p, 3]==1, 2, Mod[p, 3]==2, 0, True, a[p^e]], {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Oct 18 2018 *)

{a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 - x + 1)%n==0))}; \\ Nov 15 2002

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 + X, if( p%3==2, 1, (1 + X) / (1 - X)))) [n])}; \\ Nov 15 2002

Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3). - David W. Wilson, Aug 01 2001
a(A226946(n)) = 0; a(A034017(n)) > 0. - Reinhard Zumkeller, Jun 23 2013
a(2*n) = a(3*n + 2) = a(9*n) = a(9*n + 6) = 0. - Michael Somos, Aug 14 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*sqrt(3)/(3*Pi) = 0.367552... (A165952). - Amiram Eldar, Oct 11 2022

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.

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.

A270230 Decimal expansion of 3/(4*Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Mar 13 2016

Consider generic prisms with triangular bases (tp), enclosed by a sphere, and let f(tp) be the fraction of the sphere volume occupied by any of them (i.e., the ratio of the prism volume to the sphere volume). Then this constant is the supremum of f(tp). It is attained by prisms which have as their base equilateral triangles with edge lengths r*sqrt(2), and rectangular side faces that are r*sqrt(2) wide and r*2/sqrt(3) high, where r is the radius of the enclosing, circumscribed sphere.
An intriguing fact is that the volume of such a best-fitting prism is exactly r^3. Hence, 1/a is the volume of a sphere with radius 1.
Examples of similar constants obtained for other shapes enclosed by spheres are: A020760 for cylinders and A165952 for cuboids.

			0.238732414637843003653325645058771543051689468610684673121501016...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A002193, A019699 (one tenth of 1/a), A020760, A020832 (one tenth of 2/sqrt(3)), A165952.

First@ RealDigits[N[3/4/Pi, 120]] (* Michael De Vlieger, Mar 15 2016 *)

PARI
```
3/4/Pi
```

cp's OEIS Frontend

A049541 Decimal expansion of 1/Pi.

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A020760 Decimal expansion of 1/sqrt(3).

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A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

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

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

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Mathematica

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

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A270230 Decimal expansion of 3/(4*Pi).

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

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