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

0,1

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

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

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.

Formula

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

Example

0.70980150661400782746374731464451797194994085344545...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A021858 A368009 A368497 * A014565 A073115 A176444

Adjacent sequences: A348677 A348678 A348679 * A348681 A348682 A348683

Keyword

nonn,cons

Author

Amiram Eldar, Oct 29 2021

Status

approved