A348680 Decimal expansion of the average length of a chord in a unit square defined by a point on the perimeter and a direction, both uniformly and independently chosen at random.

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

Offset: 0

Author

Amiram Eldar, Oct 29 2021

Keywords

• nonn
• cons

Examples

			0.70980150661400782746374731464451797194994085344545...
		

References

• A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 221, ex. 2.3.7.

Links

• Rodney Coleman, Random paths through convex bodies, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; alternative link; author's link.
• Maurice Horowitz, Probability of random paths across elementary geometrical shapes, Journal of Applied Probability, Vol. 2, No. 1 (1965), pp. 169-177; Correction, ibid., Vol. 3, No. 1 (1966), p. 285.
• Philip W. Kuchel and Rodney J. Vaughan, Average Lengths of Chords in a Square, Mathematics Magazine, Vol. 54, No. 5 (1981), pp. 261-269.

Crossrefs

Cf. A091505, A247674, A348681, A348682, A348683.

Programs

Mathematica

RealDigits[(3 * Log[1 + Sqrt[2]] + 1 - Sqrt[2])/Pi, 10, 100][[1]]

PARI

(3*log(1 + sqrt(2)) + 1 - sqrt(2))/Pi \\ Michel Marcus, Oct 29 2021

Formula

Equals (3*log(1 + sqrt(2)) + 1 - sqrt(2))/Pi.