

A060294


Decimal expansion of Buffon's constant 2/Pi.


45



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

0,1


COMMENTS

The probability P(l,d) that a needle of length l will land on a line, given a floor with equally spaced parallel lines at a distance d (>=l) apart, is (2/Pi)*(l/d).  Benoit Cloitre, Oct 14 2002
Lim_{n>infinity} z(n)/log(n) = 2/Pi, where z(n) is the expected number of real zeros of a random polynomial of degree n with real coefficients chosen from a standard Gaussian distribution (cf. Finch reference).  Benoit Cloitre, Nov 02 2003
Also the ratio of the average chord length when two points are chosen at random on a circle of radius r to the maximum possible chord length (i.e., diameter) = A088538*r / 2*r = 2/Pi. Is there a (direct or obvious) relationship between this fact and that 2/Pi is the "magic geometric constant" for a circle (see MathWorld link)?  Rick L. Shepherd, Jun 22 2006
Blatner (1997) says that Euler found a "fascinating infinite product" for Pi involving the prime numbers, but the number he then describes does not match Pi. Switching the numerator and the denominator results in this number.  Alonso del Arte, May 16 2012
2/Pi is also the height (the ordinate y) of the geometric centroid of each arbelos (see the references and links given under A221918) with a large radius r=1 and any small ones r1 and r2 = 1  r1, for 0 < r1 < 1. Use the integral formula given, e.g., in the MathWorld or Wikipedia centroid reference, for the two parts of the arbelos (dissected by the vertical line x = 2*r1), and then use the decomposition formula. The heights y1 and y2 of the centroids of the two parts satisfy: F1(r1)*y1(r1) = 2*r1^2*(1r1) and F2(1r1)*y2(1r1) = 2*(1r1)^2*r1. The r1 dependent area F = F1 + F2 is Pi*r1*(1r1). (F1 and F2 are rather complicated but their explicit formulas are not needed here.) The r1 dependent horizontal coordinate x with origin at the left tip of the arbelos is x = r1 + 1/2.  Wolfdieter Lang, Feb 28 2013
Construct a quadrilateral of maximal area inside a circle. The quadrilateral is necessarily an inscribed square (with diagonals that are diameters). 2/Pi is the ratio of the square's area to the circle's area.  Rick L. Shepherd, Aug 02 2014
The expected number of real roots of a real polynomial of degree n varies as this constant times the (natural) logarithm of n, see Kac, when its coefficients are chosen from the standard uniform distribution. This may be related to Rick Shepherd's comment.  Charles R Greathouse IV, Oct 06 2014
2/Pi is also the minimum value, at x = 1/2, on (0,1) of 1/(Pi*sqrt(x*(1x))), the nonzero piece of the probability density function for the standard arcsine distribution.  Rick L. Shepherd, Dec 05 2016
The average distance from the center of a unitradius circle to the midpoints of chords drawn between two points that are uniformly and independently chosen at random on the circumference of the circle.  Amiram Eldar, Sep 08 2020


REFERENCES

David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.
G. Buffon, Essai d'arithmétique morale. Supplément à l'Histoire Naturelle, Vol. 4, 1777.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 141
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 7, eq. (1.2) and p. 105 eq. (7.4.2) with s=1/2.
Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, 1991.
Daniel A. Klain and GianCarlo Rota, Introduction to Geometric Probability, Cambridge, 1997, see Chap. 1.
Luis A. Santaló, Integral Geometry and Geometric Probability, AddisonWesley, 1976.


LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000
K. S. Brown, MathPages: The Algebra of an Infinite Grid of Resistors
G. Buffon, Essai d'arithmétique morale, Supplément à l'Histoire Naturelle, Vol. 4, 1777.
Encyclopedia of Mathematics, Arcsine distribution
Boris Gourevitch, L'univers de Pi
Mark Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:4 (1943), pp. 314320.
Veikko Nevanlinna, On constants connected with the prime number theorem for arithmetic progressions, Annales Academiae Scientiarum Fennicae Ser. A. I., No. 539 (1973).
Herbert Solomon, Geometric Probability, SIAM, 1978, p. 152. [See average chord length comment]
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Magic Geometric Constants.
Eric Weisstein's World of Mathematics, Prime Products.
Eric Weisstein's World of Mathematics, Geometric Centroid.
Wikipedia, Centroid.
Index entries for transcendental numbers


FORMULA

2/Pi = 1  5*(1/2)^3 + 9*((1*3)/(2*4))^3  13*((1*3*5)/(2*4*6))^3 ...  Jason Earls [formula corrected by Paul D. Hanna, Mar 23 2013]
The preceding formula is 2/Pi = Sum_{n>=0} (1)^n * (4*n+1) * Product_{k=1..n} (2*k1)^3/(2*k)^3.  Alexander R. Povolotsky, Mar 24 2013. [See the Hardy reference.  Wolfdieter Lang, Nov 13 2016]
2/Pi = Product_{n>=2} (p(n) + 2  (p(n) mod 4))/p(n), where p(n) is the nth prime.  Alonso del Arte, May 16 2012
2/Pi = Sum_{k>=0} ((2*k)!/(k!)^2)^3*((42*k+5)/(2^{12*k+3})) (due to Ramanujan).  L. Edson Jeffery, Mar 23 2013
Equals sinc(Pi/2).  Peter Luschny, Oct 04 2019
From A.H.M. Smeets, Apr 11 2020: (Start)
Equals Product_{i > 0} cos(Pi/2^(i+1)).
Equals Product_{i > 0} f_i(2)/2, where f_0(2) = 0, f_(i+1)(2) = sqrt(2+f_i(2)) for i >= 0; a formula by François Viète (16th century).
Note that cos(Pi/2^(i+1)) = f_i(2)/2, i >= 0. (End)
Equals lim_{n>infinity} (1/n) * Sum_{k=1..n} abs(sin(k * m)) for all nonzero integers m (conjectured). Works with cos also.  Dimitri Papadopoulos, Jul 17 2020
From Amiram Eldar, Sep 08 2020: (Start)
Equals Product_{k>=1} (1  1/(2*k)^2).
Equals lim_{k>oo} (2*k+1)*binomial(2*k,k)^2/2^(4*k).
Equals Sum_{k>=0} binomial{2*k,k)^2/((2*k+2)*2^(4*k)). (End)
Equals Sum_{k>=0} mu(4*k+1)/(4*k+1) (Nevanlinna, 1973).  Amiram Eldar, Dec 21 2020


EXAMPLE

2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...


MAPLE

Digits:=100: evalf(2/Pi); # Wesley Ivan Hurt, Aug 02 2014


MATHEMATICA

RealDigits[ N[ 2/Pi, 111]][[1]]


PROG

(PARI) default(realprecision, 20080); x=20/Pi; for (n=0, 20000, d=floor(x); x=(xd)*10; write("b060294.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009
(MAGMA) R:= RealField(100); 2/Pi(R); // G. C. Greubel, Mar 09 2018


CROSSREFS

Cf. A000796 (Pi), A088538, A154956, A082542 (numerators in an infinite product), A053300 (continued fraction without the initial 0).
Cf. A076668 (sqrt(2/Pi)).
Sequence in context: A229522 A227400 A137245 * A181171 A193025 A021615
Adjacent sequences: A060291 A060292 A060293 * A060295 A060296 A060297


KEYWORD

cons,nonn


AUTHOR

Jason Earls, Mar 28 2001


STATUS

approved



