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Decimal expansion of 1/Pi.
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%I #153 Aug 21 2024 05:41:53

%S 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,

%T 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,

%U 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

%N Decimal expansion of 1/Pi.

%C 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

%C 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

%D J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Baasel, p. 245. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.

%D C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London, 1945, p. 53, Gauge Points.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

%H G. C. Greubel, <a href="/A049541/b049541.txt">Table of n, a(n) for n = 0..10000</a>

%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/27646572">An Expression for Pi, Problem #870</a>, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. <a href="http://www.jstor.org/stable/27646723">Solution</a> appeared in Vol. 40, No. 1, January 2009, pp. 62-64.

%H J. Bohr, <a href="http://web.archive.org/web/20061003064200/http://ic.net/~jnbohr/java/Ramanujan.html">Ramanujan's Method of Approximating Pi</a>.

%H J. Borwein, <a href="http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/node15.html#SECTION00080000000000000000">Ramanujan's Sum</a>.

%H Heng Huat Chan, Shaun Cooper, and Wen-Chin Liaw, <a href="http://dx.doi.org/10.1090/S0002-9939-07-09031-4">The Rogers-Ramanujan continued fraction and a quintic iteration for 1/Pi</a>, Proc. Amer. Math. Soc. 135 (2007), 3417-3424.

%H D. V. Chudnovsky and G. V. Chudnovsky, <a href="https://doi.org/10.1073/pnas.86.21.8178">The computation of classical constants</a>, Proc. Nati. Acad. Sci. USA, Vol. 86, pp. 8178-8182, November 1989.

%H J. Guillera, <a href="http://dx.doi.org/10.1080/10586458.2006.10128943">A New Method to Obtain Series for 1/Pi and 1/Pi^2</a>, Experimental Mathematics, Volume 15, Issue 1, 2006.

%H R. Matsumoto, <a href="http://babel.altavista.com/translate.dyn?url=http://www.pluto.ai.kyutech.ac.jp/plt/matumoto/pi_small/node26.html&amp;lp=ja_en">Ramanujan Type Series</a>. [Broken link]

%H A. S. Nimbran, <a href="http://www.indianmathsociety.org.in">Deriving Forsyth-Glaisher type series for 1/π and Catalan's constant by an elementary method</a>, The Mathematics Student, Indian Math. Soc., Vol. 84, Nos. 1-2, Jan.-June (2015), 69-86. [Broken link]

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Octahedron.html">Octahedron</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F 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

%F 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

%F 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

%F 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

%F Equal Sum_{k>=2} tan(Pi/2^k)/2^k. - _Amiram Eldar_, Aug 05 2020

%F 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

%e 0.3183098861837906715377675267450287240689192914809128974953...

%p Digits:=100: evalf(1/Pi); # _Wesley Ivan Hurt_, Aug 29 2016

%t RealDigits[N[1/Pi, 10, 100]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 18 2009 *)

%o (PARI) 1/Pi \\ _Charles R Greathouse IV_, Jun 16 2011

%o (MATLAB) 1/pi \\ _Altug Alkan_, Apr 10 2016

%o (Magma) R:= RealField(100); 1/Pi(R); // _G. C. Greubel_, Aug 21 2018

%Y Cf. A000796, A165922, A165952, A165953, A165954, A063723, A010503. - _Rick L. Shepherd_, Oct 01 2009

%Y Cf. A088538 (4/Pi).

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_, Dec 11 1999