A089491 - OEIS

A089491

Decimal expansion of Buffon's constant 3/Pi.

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

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OFFSET

0,1

COMMENTS

Whereas 2/Pi (A060294) is the probability that a needle will land on one of many parallel lines, this is the probability that a needle will land on one of many lines making up a grid.

The probability that the boundary of an equilateral triangle will intersect one of the parallel lines if the triangle edge length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=3d.) - Rick L. Shepherd, Jan 11 2006

Related grid problems are discussed in the Weisstein/MathWorld Buffon-Laplace Needle Problem link. - Rick L. Shepherd, Jan 11 2006

The area of a regular dodecagon circumscribed in a unit-area circle. - Amiram Eldar, Nov 05 2020

From Bernard Schott, Apr 19 2022: (Start)

For any non-obtuse triangle ABC (see Mitrinović and Oppenheim links):

(a/A + b/B + c/C)/(a+b+c) >= 3/Pi,

(a^2/A + b^2/B + c^2/C)/(a^2+b^2+c^2) <= 3/Pi,

where (A,B,C) are the angles (measured in radians) and (a,b,c) the side lengths of this triangle.

Equality stands iff triangle ABC is equilateral. (End)

Value of the Eisenstein series E_2(z) = 1 - 24*Sum_{n>=1} sigma(n)*q (cf. A006352) at z = i. This can be seen from the relation E_2((a*z+b)/(c*z+d)) = (c*z+d)^2*E_2(z) - I*(6/Pi)*c*(c*z+d). See Proposition 6 in the link of D. Zagier for a proof. - Jianing Song, Mar 24 2026

REFERENCES

Joe Portney, Portney's Ponderables, Litton Systems, Inc., Appendix 2, 'Buffon's Needle' by Lawrence R. Weill, 200, pp. 135-138.

LINKS

Table of n, a(n) for n=0..104.

Harry Khamis, Buffon's Needle Problem.

D. S. Mitrinović, J. E. Pečarić, and V. Volenec, Recent Advances In Geometric Inequalities, Kluwer Academic Publishers, 1989, Inequalities 4.11, p. 170.

A. Oppenheim, Problem E 2649, American Mathematical Monthly, 84 (1977), p. 294.

Kevin Peterson, A Problem in Geometric Probability: Buffon's Needle Problem.

George Reese, Buffon's Needle, An Analysis and Simulation.

Shodor Education Foundation, Inc., Buffon's needle.

Washington and Lee University, Problem 18: Buffon's Needle Again. [Broken link]

Eric Weisstein's World of Mathematics, Buffon's needle problem.

Eric Weisstein's World of Mathematics, Buffon-Laplace needle problem.

Eric Weisstein's World of Mathematics, Generalized Diameter.

D. Zagier, Eisenstein Series and the Discriminant Function.

Index entries for transcendental numbers

FORMULA

Equals sinc(Pi/6). - Peter Luschny, Oct 04 2019