A019699 -id:A019699 - cp's OEIS frontend

A019692 Decimal expansion of 2*Pi.

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

Offset: 1

N. J. A. Sloane

Pi/5 or 2*Pi/10 is the expected surface area containing completely a Brownian curve (trajectory) on a plane. - Lekraj Beedassy, Jul 28 2005
Bob Palais considers this a more fundamental constant than Pi, see the Palais reference and link. - Jonathan Vos Post, Sep 10 2010
The Persian mathematician Jamshid al-Kashi seems to have been the first to use the circumference divided by the radius as the circle constant. In Treatise on the Circumference published 1424 he calculated the circumference of a unit circle to 9 sexagesimal places. - Peter Harremoës, John W. Nicholson, Aug 02 2012
"Proponents of a new mathematical constant tau (τ), equal to two times π, have argued that a constant based on the ratio of a circle's circumference to its radius rather than to its diameter would be more natural and would simplify many formulas" (from Wikipedia). - Jonathan Sondow, Aug 15 2012
The constant 2*Pi appears in the formula for the period T of a simple gravity pendulum. For small angles this period is given by Christiaan Huygens’s law, i.e., T = 2*Pi*sqrt(L/g), see for more information A223067. - Johannes W. Meijer, Mar 14 2013
There are seven consecutive nines at positions 762 to 768. - Roland Kneer, Jul 05 2013
Volume of a cylinder in which a sphere of radius 1 can be inscribed. - Omar E. Pol, Sep 25 2013
2*Pi is also the surface area of a sphere whose diameter equals the square root of 2. More generally, x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 18 2013
From Bernard Schott, Jan 31 2020: (Start)
Also, (2*Pi)*a^2 is the area of the deltoid (an hypocycloid with three cusps) whose Cartesian parametrization is:
   x = a * (2*cos(t) + cos(2*t)),
   y = a * (2*sin(t) - sin(2*t)).
The length of this deltoid is 16*a. See the curve at the Mathcurve link. (End)
Pi/5 = 0.1 * 2*Pi is the mean area of the plane triangles formed by 3 points independently and uniformly chosen at random on the surface of a unit-radius sphere. - Amiram Eldar, Aug 06 2020

			6.283185307179586476925286766559005768394338798750211641949889184615632...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4, p. 17.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 69.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Robert Ferréol, Deltoid, Mathcurve.
Christophe Garban and José A. Trujillo Ferreras, The expected area of the filled planar Brownian loop is pi/5, Communications in mathematical physics, Vol. 264, No. 3 (2006), pp. 797-810, preprint, arXiv:math/0504496 [math.PR], 2005.
Peter Harremoës, Al-Kashi’s constant
Michael Hartl, The Tau Manifesto.
Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Bob Palais, Web page about "Pi is wrong!".
Bob Palais, Pi is wrong!, The Mathematical Intelligencer Volume 23, Number 3, 2001, pp. 7-8.
Grant Sanderson, How pi was almost 6.283185..., 3Blue1Brown video (2018).
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 318.
Wikipedia, Tau proposals.
Wikipedia, Bailey-Borwein-Plouffe formula.
Index entries for transcendental numbers.

Cf. A058291 (continued fraction).
Cf. A000796, A019693, A019699, A091925, A244979.
Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid).

using Nemo
RR = RealField(334)
tau = const_pi(RR) + const_pi(RR)
tau |> println # Peter Luschny, Mar 14 2018

R:= RealField(100); 2*Pi(R); // G. C. Greubel, Mar 08 2018

RealDigits[N[2 Pi, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 20080); x=2*Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019692.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

# Use some guard digits when computing.
# BBP formula P(1, 16, 8, (0, 8, 4, 4, 0, 0, -1, 0)).
from decimal import Decimal as dec, getcontext
def BBPtau(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(16)
    for k in range(n):
        ek = dec(8 * k)
        s += f * ( dec(8) / (ek + 2) + dec(4) / (ek + 3)
                 + dec(4) / (ek + 4) - dec(1) / (ek + 7))
        f /= g
    return s
print(BBPtau(200))  # Peter Luschny, Nov 03 2023

e^(Zeta'(0)/Zeta(0)) = 2*Pi. - Peter Luschny, Jun 17 2018
From Peter Bala, Oct 30 2019: (Start)
2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/6) + 1/(n + 5/6) ).
2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/10) - 1/(n + 3/10) - 1/(n + 7/10) + 1/(n + 9/10) ). Cf. A091925 and A244979. (End)
From Amiram Eldar, Aug 06 2020: (Start)
Equals Gamma(1/6)*Gamma(5/6).
Equals Integral_{x=0..oo} log(1 + 1/x^6) dx.
Equals Integral_{x=0..oo} log(1 + 4/x^2) dx.
Equals Integral_{x=-oo..oo} exp(x/6)/(exp(x) + 1) dx.
Equals Sum_{k>=0} 1/((k + 1/4)*(k + 3/4)). (End)
Equals 4*Integral_{x=0..1} 1/sqrt(1 - x^2) dx (see Finch). - Stefano Spezia, Oct 19 2024

A019670 Decimal expansion of Pi/3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 11 1996

With an offset of zero, also the decimal expansion of Pi/30 ~ 0.104719... which is the average arithmetic area  of the 0-winding sectors enclosed by a closed Brownian planar path, of a given length t, according to Desbois, p. 1. - Jonathan Vos Post, Jan 23 2011
Polar angle (or apex angle) of the cone that subtends exactly one quarter of the full solid angle. See comments in A238238. - Stanislav Sykora, Jun 07 2014
60 degrees in radians. - M. F. Hasler, Jul 08 2016
Volume of a quarter sphere of radius 1. - Omar E. Pol, Aug 17 2019
Also smallest positive zero of Sum_{k>=1} cos(k*x)/k = -log(2*|sin(x/2)|). Proof of this identity: Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i = sqrt(-1). - Jianing Song, Nov 09 2019
The area of a circle circumscribing a unit-area regular dodecagon. - Amiram Eldar, Nov 05 2020

			Pi/3 = 1.04719755119659774615421446109316762806572313312503527365831486...
From _Peter Bala_, Nov 16 2016: (Start)
Case n = 1. Pi/3 = 18 * Sum_{k >= 0} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ).
Using the methods of Borwein et al. we can find the following asymptotic expansion for the tails of this series: for N divisible by 6 there holds Sum_{k >= N/6} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ) ~ 1/N^3 + 6/N^5 + 1671/N ^7 - 241604/N^9 + ..., where the sequence [1, 0, 6, 0, 1671, 0, -241604, 0, ...] is the sequence of coefficients in the expansion of ((1/18)*cosh(2*x)/cosh(3*x)) * sinh(3*x)^2 = x^2/2! + 6*x^4/4! + 1671*x^6/6! - 241604*x^8/8! + .... Cf. A024235, A278080 and A278195. (End)

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.3, p. 489.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016.
J. M. Borwein, P. B. Borwein, and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Jean Desbois and Stéphane Ouvry, Algebraic and arithmetic area for m planar Brownian paths, arXiv:1101.4135 [math-ph], Jan 21, 2011.
Index entries for transcendental numbers.

Cf. A000796 (Pi), A019669 (Pi/2), A019692 (2*Pi), A122952 (3*Pi), A003881(Pi/4), A137914, A238238, A002388, A091925, A093954, A016910, A019694, A136017, A352324.

Integral_{x=0..oo} 1/(1+x^m) dx: A013661 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), this sequence (m=6), A352125 (m=8), A094888 (m=10).

RealDigits[N[Pi/3,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

Pi/3 \\ Charles R Greathouse IV, Sep 08 2011

N=150; default(realprecision, 3+N); digits(Pi*10^N\3) \\ M. F. Hasler, Jul 08 2016

A third of A000796, a sixth of A019692, the square root of A100044.
Sum_{k >= 0} (-1)^k/(6k+1) + (-1)^k/(6k+5). - Charles R Greathouse IV, Sep 08 2011
Product_{k >= 1}(1-(6k)^(-2))^(-1). - Fred Daniel Kline, May 30 2013
From Peter Bala, Feb 05 2015: (Start)
Pi/3 = Sum {k >= 0} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k = 2F1(1/2,1/2;3/2;1/4). Similar series expansions hold for Pi^2 (A002388), Pi^3 (A091925) and Pi/(2*sqrt(2)) (A093954.)
The integer sequences A(n) := 4^n*(2*n + 1)! and B(n) := A(n)*( Sum {k = 0..n} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k ) both satisfy the second-order recurrence equation u(n) = (20*n^2 + 4*n + 1)*u(n-1) - 8*(n - 1)*(2*n - 1)^3*u(n-2). From this observation we can obtain the continued fraction expansion Pi/3 = 1 + 1/(24 - 8*3^3/(89 - 8*2*5^3/(193 - 8*3*7^3/(337 - ... - 8*(n - 1)*(2*n - 1)^3/((20*n^2 + 4*n + 1) - ... ))))). Cf. A002388 and A093954. (End)
Equals Sum_{k >= 1} arctan(sqrt(3)*L(2k)/L(4k)) where L=A000032. See also A005248 and A056854. - Michel Marcus, Mar 29 2016
Equals Product_{n >= 1} A016910(n) / A136017(n). - Fred Daniel Kline, Jun 09 2016
Equals Integral_{x=-oo..oo} sech(x)/3 dx. - Ilya Gutkovskiy, Jun 09 2016
From Peter Bala, Nov 16 2016: (Start)
Euler's series transformation applied to the series representation Pi/3 = Sum_{k >= 0} (-1)^k/(6*k + 1) + (-1)^k/(6*k + 5) given above by Greathouse produces the faster converging series Pi/3 = (1/2) * Sum_{n >= 0} 3^n*n!*( 1/(Product_{k = 0..n} (6*k + 1)) + 1/(Product_{k = 0..n} (6*k + 5)) ).
The series given above by Greathouse is the case n = 0 of the more general result Pi/3 = 9^n*(2*n)! * Sum_{k >= 0} (-1)^(k+n)*( 1/(Product_{j = -n..n} (6*k + 1 + 6*j)) + 1/(Product_{j = -n..n} (6*k + 5 + 6*j)) ) for n = 0,1,2,.... Cf. A003881. See the example section for notes on the case n = 1.(End)
Equals Product_{p>=5, p prime} p/sqrt(p^2-1). - Dimitris Valianatos, May 13 2017
Equals A019699/4 or A019693/2. - Omar E. Pol, Aug 17 2019
Equals Integral_{x >= 0} (sin(x)/x)^4 = 1/2 + Sum_{n >= 0} (sin(n)/n)^4, by the Abel-Plana formula. - Peter Bala, Nov 05 2019
Equals Integral_{x=0..oo} 1/(1 + x^6) dx. - Bernard Schott, Mar 12 2022
Pi/3 = -Sum_{n >= 1} i/(n*P(n, 1/sqrt(-3))*P(n-1, 1/sqrt(-3))), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The first twenty terms of the series gives the approximation Pi/3 = 1.04719755(06...) correct to 8 decimal places. - Peter Bala, Mar 16 2024
Equals Integral_{x >= 0} (2*x^2 + 1)/((x^2 + 1)*(4*x^2 + 1)) dx. - Peter Bala, Feb 12 2025

A102753 Decimal expansion of (Pi^2)/2.

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005

Also equals the area under the peak-shaped even function f(x)=x/sinh(x).
Proof: For the upper half of the integral, write f(x) = 2x*exp(-x)/(1-exp(-2x)) = sum_{k=1..infinity} 2x*exp(-(2k-1)x) and integrate term by term from zero to infinity. - Stanislav Sykora, Nov 01 2013
Volume of the 4-dimensional unit sphere; the volume of the n-dimensional unit sphere is Pi^(n/2)/gamma(n/2+1) (see n-ball link and A164103). - Rick L. Shepherd, Jun 22 2017
Pi^2/2 is the squared side-length of a square with diagonal Pi. - Wesley Ivan Hurt, Jan 28 2022

			4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....

J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.

G. C. Greubel, Table of n, a(n) for n = 1..10000
T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, equation 2.39, arXiv:0705.2379 [math.CA], 2007.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Hypersphere.
Eric Weisstein's World of Mathematics, Trigamma Function.
Wikipedia, Hypersphere.
Wikipedia, Volume of an n-ball.
Index entries for transcendental numbers

Cf. A002388, A000796, A019699, A026424, A164103, A164105, A164106, A164108, A248359, A276023.

RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)

PARI
```
Pi^2/2 \\ Michel Marcus, Sep 04 2015
```

Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).
Equals the volume of revolution of the sine or cosine curve for one half period, Integral_{0,Pi} Sin(x)^2 dx. - Robert G. Wilson v, Dec 15 2005
Equals Sum_{k >=1} 4^k/(k^2*binomial(2*k,k)) [Amdeberhan, Sprugnoli]. - R. J. Mathar, Sep 28 2007
Equals 4*Sum_{k >=1} 1/(2k-1)^2 [Wells].
From  Peter Bala, Nov 05 2019: (Start)
Pi^2/2 = Integral_{x = 0..inf} cosh(x)*x^2/sinh(x)^2 dx.
Pi^2/2 = 5*sum_{k >= 0} binomial(2*k,k)(-1/16)^k*1/(2*k+1)^2.
Pi^2/2 = 10*Integral_{x = 0..1/2}  1/x*log(x + sqrt(1 + x^2)) dx. (End)
Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - Amiram Eldar, Aug 17 2020
Conjecture: Pi^2/2 = Sum_{n = -oo..oo} ( cos(Pi*sqrt(n^2+1)) - cos(Pi*n) ) (using the Eisenstein summation convention). - Peter Bala, Oct 08 2021
Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - Bernard Schott, Jan 28 2022

A019693 Decimal expansion of 2*Pi/3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Volume between a cylinder and the inscribed sphere of radius 1. - Omar E. Pol, Sep 25 2013
(2/3)*Pi is also the surface area of a sphere whose diameter equals the square root of 2/3. More generally, x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Jun 18 2018
Volume of a hemisphere of radius 1. - Omar E. Pol, Aug 17 2019
Angle in radians between two vectors of equal magnitude and originating from the same point whose vector sum also has the same magnitude. - Stefano Spezia, Jun 30 2025

			2.094395102393195492308428922186335256131446... - _Omar E. Pol_, Sep 25 2013

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, pp. 235-236.
Index entries for transcendental numbers.

Cf. A000796, A019692, A019699, A157697.

RealDigits[N[2*Pi/3, 6!]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

PARI
```
2*Pi/3 \\ Michel Marcus, Jun 19 2018
```

Equals 5*A019699. - Omar E. Pol, Aug 17 2019
Equals arccos(-1/2). - Amiram Eldar, Aug 12 2020
Equals Integral_{x=-oo..oo} 1/(1 + x^6) dx. - Stefano Spezia, Mar 05 2022
Equals Integral_{x=0..2*Pi} 1/(5 - 4*cos(x)) dx. - Kritsada Moomuang, May 22 2025

A232808 Decimal expansion of the surface area of a 3D sphere with unit volume.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 30 2013

More generally, the ratio (surface)/(volume)^(2/3), characteristic of the shape of a bounded 3D body, which is invariant under linear scaling and known as the surface index. Its common value for all spheres is the smallest possible among all closed 3D bodies (for a cube, for example, it is exactly 6.0).

			4.83597586204940892215090053991785481683384221697158466707687622613685...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Index entries for transcendental numbers

Cf. A000796 (Pi), A019673 (Pi/6); other sphere metrics: A019694, A019699, A087198, A087199.

RealDigits[(36 Pi)^(1/3), 10, 90][[1]] (* Bruno Berselli, Dec 01 2013 *)

(36*Pi)^(1/3) = 6*A019673^(1/3).

A374772 Decimal expansion of the upper bound of the density of sphere packing in the Euclidean 3-space resulting from the dodecahedral conjecture.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 19 2024

See A374753 for more information on the dodecahedral conjecture.

Also isoperimetric quotient (see A381671 for definition) of a regular dodecahedron. - Paolo Xausa, May 19 2025

			0.7546973993374058303916521059902293313424321921459...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Local Density.
Index entries for transcendental numbers.

Cf. A374753 (dodecahedral conjecture), A374755 (strong dodecahedral conjecture), A374771, A374837, A374838.

Cf. A093825, A019699, A102769, A131595, A381671.

First[RealDigits[Pi*Sqrt[5 + Sqrt[5]]/(15*Sqrt[10]*(Sqrt[5] - 2)), 10, 100]]

Pi*sqrt(5 + sqrt(5))/(15*sqrt(10)*(sqrt(5) - 2)) \\ Charles R Greathouse IV, Feb 07 2025

Equals (4/3)*Pi/A374753 = 10*A019699/A374753.
Equals A374771/A102769.
Equals Pi*sqrt(5 + sqrt(5))/(15*sqrt(10)*(sqrt(5) - 2)).
Equals 4*Pi/A374755.
Equals 36*Pi*A102769^2/(A131595^3). - Paolo Xausa, May 19 2025

A164106 Decimal expansion of 16*Pi^3/105.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 7-dimensional unit sphere.

			Equals 4.72476597033140116959639086736783164986290111...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164108.

RealDigits[16*Pi^3/105, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

16*Pi^3/105 \\ G. C. Greubel, Apr 09 2017

Equals 16*A091925/105 = A164107/7 .

A164108 Decimal expansion of Pi^4/24.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 8-dimensional unit sphere.

			4.0587121264167682181850138620293796354053160696952259038...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Cornel Ioan Vălean, Problem 11993, The American Mathematical Monthly, Vol. 124, No. 7 (2017), p. 659; An Integral Related to Euler Sums, Solution to Problem 11993 by Abdelhak Berkane, ibid., Vol. 126, No. 4 (2019), pp. 373-374.
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, n-sphere.
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164106.

RealDigits[Pi^4/24, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^4/24 \\ G. C. Greubel, Apr 09 2017
```

Equals A164109/8 = A092425/24 = A072691*A102753.

Pi^4/240 = -Integral_{x=0..1} log(1-x)*log(1+x)^2/x dx (Vălean, 2017). - Amiram Eldar, Mar 26 2022

A374753 Decimal expansion of the volume of a regular dodecahedron having unit inradius.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 19 2024

The dodecahedral conjecture (proved in 1988 by Thomas C. Hales and Sean McLaughlin, see links) states that, in any packing of unit spheres in the Euclidean 3-space, every Voronoi cell has volume at least equal to this value.

			5.55029102851551026907043211366183924073759821288...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Thomas C. Hales and Sean McLaughlin, A proof of the dodecahedral conjecture, arXiv:math/9811079 [math.MG], 2008.
Thomas C. Hales and Sean McLaughlin, The Dodecaheral Conjecture, Journal of the American Mathematical Society, Vol. 23, No. 2, April 2010, pp. 299-344.
Wikipedia, Dodecahedral conjecture.
Wikipedia, Regular dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A019699, A374755 (strong dodecahedral conjecture), A374772 (density).

First[RealDigits[10*Sqrt[130 - 58*Sqrt[5]], 10, 100]]

Equals (4/3)*Pi/A374772 = 10*A019699/A374772.
Equals 10*sqrt(130 - 58*sqrt(5)).
Equals A374755/3.

A164086 Beatty sequence for 4*Pi/3 = 4.1887902... .

Original entry on oeis.org

4, 8, 12, 16, 20, 25, 29, 33, 37, 41, 46, 50, 54, 58, 62, 67, 71, 75, 79, 83, 87, 92, 96, 100, 104, 108, 113, 117, 121, 125, 129, 134, 138, 142, 146, 150, 154, 159, 163, 167, 171, 175, 180, 184, 188, 192, 196, 201, 205, 209, 213, 217, 222, 226, 230, 234, 238, 242

Offset: 1

Reinhard Zumkeller, Aug 11 2009

nonn

a(n) = A109238(n) for n <= 20;
complement of A164087;
a(n) = A164088(A164087(n)) and A164088(a(n)) = A164087(n);
a(A000578(n)) = A066645(n).

			a(3^3) = a(27) = 113 = (integer part of volume of sphere with radius=3) = A066645(3).

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A022844, A019699.

With[{c=4 \[Pi]/3},Floor[c #]&/@Range[70]]  (* Harvey P. Dale, Mar 19 2011 *)

fprec:100$
A164086:4*%pi/3$
ev(bfloat(A164086)); /* Martin Ettl, Nov 03 2012 */

a(n) = floor(4*n*Pi/3).

cp's OEIS Frontend

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A102753 Decimal expansion of (Pi^2)/2.

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A019693 Decimal expansion of 2*Pi/3.

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A232808 Decimal expansion of the surface area of a 3D sphere with unit volume.

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A374772 Decimal expansion of the upper bound of the density of sphere packing in the Euclidean 3-space resulting from the dodecahedral conjecture.

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A164106 Decimal expansion of 16*Pi^3/105.