A088538 Decimal expansion of 4/Pi.

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

Offset: 1

Author

Benoit Cloitre, Nov 16 2003

Keywords

• cons
• nonn
• easy

Comments

Average length of chord formed from two randomly chosen points on the circumference of a unit circle (see Weisstein/MathWorld link). - Rick L. Shepherd, Jun 19 2006

Suppose u(0) = 1 + i where i^2 = -1 and u(n+1) = (1/2)*(u(n) + |u(n)|). Conjecture: lim_{n -> oo} Real(u(n)) = 4/Pi. - Yalcin Aktar, Jul 18 2007

Ratio of the arc length of the cycloid for one period to the circumference of the corresponding circle rolling on a line. Thus, for any integral number n of revolutions of a circle of radius r, a point on the circle travels (4/Pi)*2*Pi*r*n = 8*r*n (while the center of the circle moves only 2*Pi*r*n). This ratio varies for partial revolutions and depends upon the initial position of the point with points nearest the line moving the slowest (see Dudeney, who explains how the tops of bicycle wheels move faster than the parts nearest the ground). - Rick L. Shepherd, May 05 2014

Average distance traveled in two steps of length 1 for a random walk in the plane starting at the origin. - Jean-François Alcover, Aug 04 2014

Ratio of the circle area to the area of a square having equal perimeters. - Iaroslav V. Blagouchine, May 06 2016

This is also the value of a special case (n=1) of an n-family of series considered by Hardy (see A278145): 1 + (1/2)*(1/2)^2 + (1/3)*(1*3/(2*4))^2 + (1/4)*((1*3*5) / (2*4*6))^2 + ... = Sum_{k>=0} (1/(k+1))*((2*k-1)!!/(2*k)!!)^2. - Wolfdieter Lang, Nov 14 2016

Minimum ratio of the area of a rectangle to one of its inscribed ellipses, or only existing ratio if the rectangle is a square and then the only inscribed ellipse is a circle. This ellipse has its semiaxes parallel to the sides of the rectangle. If a rectangle has sides of length 2a and 2b, its area is 4*a*b, while the ellipse inscribed has a and b as semiaxes, therefore its area is a*b*Pi. Thus the ratio is (4*a*b)/(a*b*Pi) = 4/Pi. - Giovanni Zedda, Jun 20 2019

The diameter of the conventional spherical earth is (4/Pi)*10000 km = 12732.395... km. - Jean-François Alcover, Oct 30 2021

From Jianing Song, Aug 06 2022: (Start)

sign(sin(x)) = (4/Pi) * Sum_{n>=0} sin((2*n+1)*x)/(2*n+1), for all x in R;

sign(cos(x)) = (4/Pi) * Sum_{n>=0} (-1)^n*cos((2*n+1)*x)/(2*n+1), for all x in R. (End)

Examples

			4/Pi = 1.2732395.... = 1/0.78539816...
		

References

• Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 195.
• H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp. 99, 300-301, #294.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, p. 86.
• G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 105, eq. (7.5.1) for n=1.
• L. B. W. Jolley, Summation of Series, Dover (1961).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• J.-P. Allouche, On a formula of T. Rivoal, arXiv:1307.3906 [math.NT], 2013.
• Friedrich L. Bauer, Historische Notizen / Wallis-artige Kettenprodukte, Informatik Spektrum 31,4 (2008) 348-352.
• J. M. Borwein, A. Straub, J. Wan, and W. Zudilin, Densities of short uniform random walks, arXiv:1103.2995 [math.CA], 2011.
• Manuel Benito, Luis M. Navas and Juan Luis Varona, Möbius inversion from the point of view of arithmetical semigroup flows, Proceedings of the “Segundas Jornadas de Teoría de Números” (Madrid, 2007), 63-81
• R. J. Mathar, Chebyshev Series Expansion of Inverse Polynomials, arXiv:0403344 [math.CA], 2004-2005.
• Michael I. Shamos, A catalog of the real numbers, (2007). See p. 309.
• Eric Weisstein's World of Mathematics, Circle Line Picking.
• Eric Weisstein's World of Mathematics, Cycloid.
• Index entries for transcendental numbers.

Crossrefs

Cf. A079097 for terms of a generalized continued fraction for 4/Pi. Inverse of A003881. A060294, A278145, A049541 (1/Pi).

Programs

Mathematica

RealDigits[N[4/Pi, 6!]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI

4/Pi \\ Charles R Greathouse IV, Jun 21 2013

Formula

4/Pi = Product_(1-(-1)^((p-1)/2)/p) where p runs through the odd primes.

Arcsin x = (4/Pi) Sum_{n = 1, 3, 5, 7, ...} T_n(x)/n^2 (Chebyshev series of arcsin; App C of math.CA/0403344). - R. J. Mathar, Jun 26 2006

Equals 1 + Sum_{n >= 1} ((2n-3)!!/(2n)!!)^2. [Jolley eq 274]. - R. J. Mathar, Nov 03 2011

Equals binomial(1,1/2). - Bruno Berselli, May 17 2016

2*A060294 (twice Buffon's constant) = 1/Gamma(3/2)^2. - Wolfdieter Lang, Nov 14 2016

Equals 1 + Sum_{n>=0} (Catalan(n)/2^(2*n+1))^2, with Catalan(n) = A000108(n). This is the rewritten Jolley (274) series. See the above R. J. Mathar entry with (-1)!! := 1. - Ralf Steiner, Sep 18 2018

4/Pi = 1 + (1/4)*hypergeometric([1, 1/2, 1/2], [2, 2], 1) = hypergeometric([-1/2, -1/2], [1], 1). From the g.f. of Catalan^2 given in A001246. - Wolfdieter Lang, Sep 18 2018

Equals Product_{k>=1} (1 + 1/(4*k*(k+1))). - Amiram Eldar, Aug 05 2020

From Stefano Spezia, Oct 26 2024: (Start)

4/Pi = 1 + K_{n>=1} n^2/(2*n + 1), where K is the Gauss notation for an infinite continued fraction. In the expanded form, 4/Pi = 1 + 1^2/(3 + 2^2/(5 + 3^2/(7 + 4^2/(9 + 5^2/(11 + ...))))) (see Finch at p. 23).

4/Pi = Sum_{n>=0} tan(Pi/2^(n+2))/2^n (see Shamos). (End)

4/Pi = Sum_{n >= 0} (-1)^n * mu(2*n+1)/(2*n + 1), where mu(n) is the Möbius function A008683 (see, for example, Benito et al., p. 77). - Peter Bala, Jan 07 2025

Equals Integral_{x=0..1} (2*EllipticK(x))/Pi dx. - Kritsada Moomuang, Jun 04 2025