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A060294
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Decimal expansion of Buffon's constant 2/Pi.
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31
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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
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0,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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
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
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FORMULA
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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
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EXAMPLE
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2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...
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MATHEMATICA
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RealDigits[ N[ 2/Pi, 111]][[1]]
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PROG
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(PARI) { default(realprecision, 20080); x=20/Pi; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b060294.txt", n, " ", d)); } [From Harry J. Smith, Jul 03 2009]
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CROSSREFS
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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: A199186 A176715 A137245 * A181171 A193025 A021615
Adjacent sequences: A060291 A060292 A060293 * A060295 A060296 A060297
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KEYWORD
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cons,nonn
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AUTHOR
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Jason Earls (zevi_35711(AT)yahoo.com), Mar 28 2001
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STATUS
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approved
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