cp's OEIS Frontend

A019692 Decimal expansion of 2*Pi.

%I A019692 #129 Apr 17 2025 05:41:58
%S A019692 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,
%T A019692 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,
%U A019692 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
%N A019692 Decimal expansion of 2*Pi.
%C A019692 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
%C A019692 Bob Palais considers this a more fundamental constant than Pi, see the Palais reference and link. - _Jonathan Vos Post_, Sep 10 2010
%C A019692 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
%C A019692 "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
%C A019692 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
%C A019692 There are seven consecutive nines at positions 762 to 768. - _Roland Kneer_, Jul 05 2013
%C A019692 Volume of a cylinder in which a sphere of radius 1 can be inscribed. - _Omar E. Pol_, Sep 25 2013
%C A019692 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
%C A019692 From _Bernard Schott_, Jan 31 2020: (Start)
%C A019692 Also, (2*Pi)*a^2 is the area of the deltoid (an hypocycloid with three cusps) whose Cartesian parametrization is:
%C A019692    x = a * (2*cos(t) + cos(2*t)),
%C A019692    y = a * (2*sin(t) - sin(2*t)).
%C A019692 The length of this deltoid is 16*a. See the curve at the Mathcurve link. (End)
%C A019692 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
%D A019692 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4, p. 17.
%D A019692 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 69.
%H A019692 Harry J. Smith, <a href="/A019692/b019692.txt">Table of n, a(n) for n = 1..20000</a>
%H A019692 Robert Ferréol, <a href="https://www.mathcurve.com/courbes2d.gb/deltoid/deltoid.shtml">Deltoid</a>, Mathcurve.
%H A019692 Christophe Garban and José A. Trujillo Ferreras, <a href="https://doi.org/10.1007/s00220-006-1555-2">The expected area of the filled planar Brownian loop is pi/5</a>, Communications in mathematical physics, Vol. 264, No. 3 (2006), pp. 797-810, <a href="https://arxiv.org/abs/math/0504496">preprint</a>, arXiv:math/0504496 [math.PR], 2005.
%H A019692 Peter Harremoës, <a href="https://web.archive.org/web/20120929205311/http://www.harremoes.dk/Peter/Undervis/Turnpage/Turnpage1.html">Al-Kashi’s constant</a>
%H A019692 Michael Hartl, <a href="http://tauday.com">The Tau Manifesto</a>.
%H A019692 Melissa Larson, <a href="https://www.d.umn.edu/~jgreene/masters_reports/BBP%20Paper%20final.pdf">Verifying and discovering BBP-type formulas</a>, 2008.
%H A019692 Bob Palais, <a href="http://www.math.utah.edu/~palais/pi.html">Web page about "Pi is wrong!"</a>.
%H A019692 Bob Palais, <a href="http://dx.doi.org/10.1007/BF03026846">Pi is wrong!</a>, The Mathematical Intelligencer Volume 23, Number 3, 2001, pp. 7-8.
%H A019692 Grant Sanderson, <a href="https://www.youtube.com/watch?v=bcPTiiiYDs8">How pi was almost 6.283185...</a>, 3Blue1Brown video (2018).
%H A019692 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 318.
%H A019692 Wikipedia, <a href="https://en.wikipedia.org/wiki/Turn_(geometry)#Tau_proposals">Tau proposals</a>.
%H A019692 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula">Bailey-Borwein-Plouffe formula</a>.
%H A019692 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A019692 e^(Zeta'(0)/Zeta(0)) = 2*Pi. - _Peter Luschny_, Jun 17 2018
%F A019692 From _Peter Bala_, Oct 30 2019: (Start)
%F A019692 2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/6) + 1/(n + 5/6) ).
%F A019692 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)
%F A019692 From _Amiram Eldar_, Aug 06 2020: (Start)
%F A019692 Equals Gamma(1/6)*Gamma(5/6).
%F A019692 Equals Integral_{x=0..oo} log(1 + 1/x^6) dx.
%F A019692 Equals Integral_{x=0..oo} log(1 + 4/x^2) dx.
%F A019692 Equals Integral_{x=-oo..oo} exp(x/6)/(exp(x) + 1) dx.
%F A019692 Equals Sum_{k>=0} 1/((k + 1/4)*(k + 3/4)). (End)
%F A019692 Equals 4*Integral_{x=0..1} 1/sqrt(1 - x^2) dx (see Finch). - _Stefano Spezia_, Oct 19 2024
%e A019692 6.283185307179586476925286766559005768394338798750211641949889184615632...
%t A019692 RealDigits[N[2 Pi, 6! ]] (* _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009 *)
%o A019692 (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
%o A019692 (Magma) R:= RealField(100); 2*Pi(R); // _G. C. Greubel_, Mar 08 2018
%o A019692 (Julia)
%o A019692 using Nemo
%o A019692 RR = RealField(334)
%o A019692 tau = const_pi(RR) + const_pi(RR)
%o A019692 tau |> println # _Peter Luschny_, Mar 14 2018
%o A019692 (Python)  # Use some guard digits when computing.
%o A019692 # BBP formula P(1, 16, 8, (0, 8, 4, 4, 0, 0, -1, 0)).
%o A019692 from decimal import Decimal as dec, getcontext
%o A019692 def BBPtau(n: int) -> dec:
%o A019692     getcontext().prec = n
%o A019692     s = dec(0); f = dec(1); g = dec(16)
%o A019692     for k in range(n):
%o A019692         ek = dec(8 * k)
%o A019692         s += f * ( dec(8) / (ek + 2) + dec(4) / (ek + 3)
%o A019692                  + dec(4) / (ek + 4) - dec(1) / (ek + 7))
%o A019692         f /= g
%o A019692     return s
%o A019692 print(BBPtau(200))  # _Peter Luschny_, Nov 03 2023
%Y A019692 Cf. A058291 (continued fraction).
%Y A019692 Cf. A000796, A019693, A019699, A091925, A244979.
%Y A019692 Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid).
%K A019692 nonn,cons,easy
%O A019692 1,1
%A A019692 _N. J. A. Sloane_