A396028 Decimal expansion of the probability that Buffon's needle will not land on any line when dropped on a triangular tiling in which the altitude of each triangle equals the length of the needle.

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

Offset

0,3

Links

Table of n, a(n) for n=0..106.

A. A. Markoff (Andrey Andreyevich Markov), Wahrscheinlichkeitsrechnung, Leipzig und Berlin: B. G. Teubner, 1912, pp. 169-173.

Michael D. Perlman and Michael J. Wichura, Sharpening Buffon's Needle, The American Statistician, Vol. 29, No. 4 (1975), pp. 157-163.

Enis Siniksaran, Throwing Buffon's Needle with Mathematica, The Mathematica Journal, Vol. 11, No. 1 (2008), pp. 71-90.

J. V. Uspensky, Introduction To Mathematical Probability, Mcgraw Hill Book Co. Inc., 1937, p. 258, Problem 8.

Eric Weisstein's World of Mathematics, Buffon-Laplace Needle Problem.

Wikipedia, Buffon's needle problem.

Wikipedia, Triangular tiling.

Index entries for transcendental numbers.

Formula

Equals 3/2 - 3*(8-sqrt(3))/(4*Pi).

Equals 1/4 - A396029.

Equals 1 - A396029 - A102519 - (A240935 - 1/4).

Example

0.0036373544636000079068897132645547366345241113895031...

Mathematica

RealDigits[3/2 - 3*(8-Sqrt[3])/(4*Pi), 10, 120, -1][[1]]

Prog

(PARI) 3/2 - 3*(8-sqrt(3))/(4*Pi)

Crossrefs

Cf. A060294, A089491, this constant (0 lines), A396029 (1 line), A102519 (2 lines), A240935 - 1/4 (3 lines).

Sequence in context: A303129 A170859 A123146 * A394170 A016661 A376827

Adjacent sequences: A396025 A396026 A396027 * A396029 A396030 A396031

Keyword

nonn,cons

Author

Amiram Eldar, May 14 2026

Status

approved