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A019692 Decimal expansion of 2*Pi. 55
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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

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).

Wikipedia, Tau proposals

Index entries for transcendental numbers

FORMULA

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)

EXAMPLE

6.283185307179586476925286766559005768394338798750211641949889184615632...

MATHEMATICA

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

PROG

(PARI) 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

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

(Julia)

using Nemo

RR = RealField(334)

tau = const_pi(RR) + const_pi(RR)

tau |> println # Peter Luschny, Mar 14 2018

CROSSREFS

Cf. A058291 (continued fraction).

Cf. A000796, A019693, A019699, A091925, A244979.

Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid).

Sequence in context: A086744 A242301 A256129 * A031259 A059629 A082577

Adjacent sequences:  A019689 A019690 A019691 * A019693 A019694 A019695

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 24 21:25 EDT 2020. Contains 337322 sequences. (Running on oeis4.)