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

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)

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.

Index entries for transcendental numbers

Formula

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

From Amiram Eldar, Aug 20 2020: (Start)

Equals Product{k>=1} cos(Pi/(6*2^k)).

Equals Product{k>=1} (1 - 1/(6*k)^2). (End)

Example

3/Pi = 0.95492965855137201461330258023508617220675787444273869248600...

Mathematica

RealDigits[ N[ 3/Pi, 111]][[1]]

Prog

(PARI) 3/Pi \\ Michel Marcus, Nov 05 2020

Crossrefs

Cf. A000796 (Pi), A060294 (2/Pi).

Sequence in context: A198933 A353772 A259148 * A199792 A193960 A377522

Adjacent sequences: A089488 A089489 A089490 * A089492 A089493 A089494

Keyword

cons,nonn

Author

Robert G. Wilson v, Nov 04 2003

Status

approved