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A060294 Decimal expansion of Buffon's constant 2/Pi. 33
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 (list; constant; graph; refs; listen; history; text; internal format)
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*(1-r1) and F2(1-r1)*y2(1-r1) = 2*(1-r1)^2*r1. The r1 dependent area F = F1 + F2 is Pi*r1*(1-r1). (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

REFERENCES

David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.

G. Buffon, Essai d'arithmetique morale. Supplement a l'Histoire Naturelle, Vol. 4, 1777.

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 141

R. Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, 1991.

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, 1997, see Chap. 1.

L. A. Santalo, Integral Geometry and Geometric Probability, Addison-Wesley, 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

Boris Gourevitch, L'univers de Pi

M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:4 (1943), pp. 314-320.

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

FORMULA

2/Pi = 1 - 5*(1/2)^3 + 9*((1*3)/(2*4))^3 - 13*((1*3*5)/(2*4*6))^3 ... - Earls [formula corrected by Paul D. Hanna, Mar 23 2013]

The identity above is 2/Pi = Sum_{n>=0} (-1)^n * (4*n+1) * Product_{k=1..n} (2*k-1)^3/(2*k)^3. - Alexander R. Povolotsky, Mar 24 2013

2/Pi = product (n = 2 .. infinity) (p(n) + 2 - (p(n) mod 4))/p(n), where p(n) is the n-th prime. - Alonso del Arte, May 16 2012

2/Pi = sum_{k=0,1,...} ((2*k)!/(k!)^2)^3*((42*k+5)/(2^{12*k+3})) (due to Ramanujan). - L. Edson Jeffery, Mar 23 2013

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=(x-d)*10; write("b060294.txt", n, " ", d)); } \\ [Harry J. Smith, Jul 03 2009]

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 (zevi_35711(AT)yahoo.com), Mar 28 2001

STATUS

approved

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Last modified October 25 17:33 EDT 2014. Contains 248557 sequences.