A093828 -id:A093828 - 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

A019694 Decimal expansion of 2*Pi/5.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also, with proper offset, decimal expansion of the magnetic permeability of vacuum in SI units, mu_0 = 4*Pi*10^-7 N A^-2, an assigned metrological constant. [This exact expression for mu_0 was valid until the 2019 SI redefinition. In the New SI, mu_0 is numerically very close to that value but is determined only up to a certain error. - Andrey Zabolotskiy, May 22 2019]
Regarding these, see A003678 for general context notes, references and links. - Stanislav Sykora, Jun 16 2012
With offset 2 this is also the decimal expansion of 4*Pi, the surface area of a sphere whose diameter equals the square root of 4, hence its radius is 1. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Jan 18 2013, Oct 05 2013, Dec 22 2013
4*Pi is also the area of the domain bounded by the witch of Agnesi whose Cartesian equation is y = 8 / (x^2 + 4) and its asymptote. More generally (4*Pi) * a^2 is the area of the domain bounded by the witch of Agnesi whose Cartesian equation is y = (8*a^3) / (x^2 + 4*a^2) and its asymptote (Eric Weisstein's link, formula 6). - Bernard Schott, Jun 28 2023

			1.2566370614359172953850573533118....
mu_0 = 12.566370614359172953850573533118... 10^-7 N/A^2. - _Stanislav Sykora_, Jun 16 2012

Ivan Panchenko, Table of n, a(n) for n = 1..1000
NIST, magnetic constant mu_0.
Nobelprize.org, The Discovery of Quasicrystals (PDF).
Eric Weisstein's World of Mathematics, Witch of Agnesi.
Wikipedia, Crystallographic Restriction Theorem.
Index entries for sequences related to curves.
Index entries for transcendental numbers.

Other assigned constants: A003678, A072915, A081799, A182999, A213610, A213611, A213612, A213613, A213614.

Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid), A336266 (double egg), A336308 (ovoid).

Digits:=100: evalf(2*Pi/5); # Wesley Ivan Hurt, Jan 07 2017

RealDigits[ 2*Pi/5, 10, 111][[1]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 and modified by Robert G. Wilson v *)

PARI
```
2*Pi/5 \\ G. C. Greubel, Sep 11 2017
```

Equals Sum_{k>=1} sin(Pi*k/5)/k. - Amiram Eldar, Aug 12 2020

Equals -zeta(3/2)/(10*zeta(-1/2)). - Mats Granvik, May 28 2022

A122952 Decimal expansion of 3*Pi.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 30 2006

Area of the unit cycloid with cusp at the origin, whose parametric formula is x = t - sin(t) and y = 1 - cos(t).
The arc length Integral_{theta=0..2*Pi} sqrt(2(1-cos(theta))) (d theta) = 8.
3*Pi is also the surface area of a sphere whose diameter equals the square root of 3. 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
3*Pi is also the area of the nephroid (an epicycloid with two cusps) whose Cartesian parametrization is: x = (1/2) * (3*cos(t) - cos(3t)) and y = (1/2) * (3*sin(t) - sin(3t)). The length of this nephroid is 12. See the curve at the Mathcurve link. - Bernard Schott, Feb 01 2020

			9.424777960769379715387930149838508652591508198125317462924833776...

Anton, Bivens & Davis, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., NY 2002, p. 490.
William H. Beyer, Editor, CRC St'd Math. Tables, 27th Edition, CRC Press, Inc., Boca Raton, FL, 1984, p. 214.

Robert Ferréol, Nephroid, Mathcurve.
Eric Weisstein's World of Mathematics, Cycloid.
Index entries for transcendental numbers

Cf. A000796, A019692, A019694, A019669.

Cf. A093828, A180434, A197723.

Mathematica
```
RealDigits[3Pi, 10, 111][[1]]
```

3*Pi \\ Charles R Greathouse IV, Sep 28 2022

A161685 Continued fraction for (3*Pi)/8.

Original entry on oeis.org

1, 5, 1, 1, 1, 1, 2, 12, 12, 6, 1, 1, 1, 1, 11, 1, 4, 1, 3, 2, 1, 6, 1, 4, 4, 3, 3, 4, 10, 1, 4, 37, 1, 1, 20, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 1, 2, 2, 15, 2, 1, 2, 2, 1, 1, 1, 3, 1, 4, 3, 83, 9, 19, 1, 2, 2, 4, 1, 1, 1, 4, 3, 4, 1, 1, 8, 2, 1, 2, 1, 7, 3, 2, 1, 1, 5, 1, 6, 1, 5, 1, 7, 17, 9, 6, 3, 1, 1, 1, 1, 2

Offset: 0

Harry J. Smith, Jun 18 2009

			1.178097245096172464423491268... = 1 + 1/(5 + 1/(1 + 1/(1 + 1/(1 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A195909, A195913, A195697. - Mohammad K. Azarian, Oct 11 2011

Cf. A093828 (decimal expansion).

SetDefaultRealField(RealField(105)); R:= RealField(); ContinuedFraction(3*Pi(R)/8); // G. C. Greubel, Aug 11 2019

convert(3*Pi/8, confrac, 105); # G. C. Greubel, Aug 11 2019

ContinuedFraction[(3*Pi)/8,120] (* Harvey P. Dale, Mar 13 2016 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(3*Pi/8); for (n=0, 20000, write("b161685.txt", n, " ", x[n+1])); }

continued_fraction_list(3*pi/8, nterms=105) # G. C. Greubel, Aug 11 2019

A162540 a(n) = (2n+1)(2n+3)(2*n+5)/3.

Original entry on oeis.org

5, 35, 105, 231, 429, 715, 1105, 1615, 2261, 3059, 4025, 5175, 6525, 8091, 9889, 11935, 14245, 16835, 19721, 22919, 26445, 30315, 34545, 39151, 44149, 49555, 55385, 61655, 68381, 75579, 83265, 91455, 100165, 109411, 119209, 129575, 140525, 152075, 164241

Offset: 0

Jacob Landon (jacoblandon(AT)aol.com), Jul 05 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A061550, A077415, A093828.

[(2*n+1)*(2*n+3)*(2*n+5)/3: n in [0..40]]; // Vincenzo Librandi, Nov 16 2011

A162540:=n->(2*n+1)*(2*n+3)*(2*n+5)/3: seq(A162540(n), n=0..80); # Wesley Ivan Hurt, May 28 2016

Table[((2n+1)(2n+3)(2n+5))/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{5,35,105,231},40] (* Harvey P. Dale, Nov 06 2011 *)

Vec((5+15*x-5*x^2+x^3)/(x-1)^4 + O(x^100)) \\ Altug Alkan, Oct 26 2015

a(n) = A061550(n)/3 = A077415(2*n+3).
From R. J. Mathar, Jul 16 2009: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: (5 + 15*x - 5*x^2 + x^3)/(x-1)^4. (End)
a(n) = 5*Pochhammer(7/2,n)/Pochhammer(1/2,n). Hence e.g.f. is 5* 1F1(7/2;1/2;x), with 1F1 being the confluent hypergemetric function (also known as Kummer's). - Stanislav Sykora, May 26 2016
E.g.f.: (8*x^3 + 60*x^2 + 90*x + 15)*exp(x)/3. - Robert Israel, May 27 2016
From Amiram Eldar, Jan 09 2021: (Start)
Sum_{n>=0} 1/a(n) = 1/4.
Sum_{n>=0} (-1)^n/a(n) = 3*Pi/8 - 1 = A093828 - 1. (End)

Offset corrected, definition clarified by R. J. Mathar, Jul 16 2009

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A019692 Decimal expansion of 2*Pi.

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A019694 Decimal expansion of 2*Pi/5.

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A122952 Decimal expansion of 3*Pi.

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A161685 Continued fraction for (3*Pi)/8.

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A162540 a(n) = (2*n+1)*(2*n+3)*(2*n+5)/3.

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A162540 a(n) = (2n+1)(2n+3)(2*n+5)/3.