A135707 Consider a domino formed from two adjacent 1 X 1 squares. This is the decimal expansion of the average distance between a random point in the left square and a random point in the right square.

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

Offset

1,3

Comments

The problem of calculating this average was apparently first published by Hans Jörg Oser in 1975, at the suggestion of the American mathematician Charles Royal Johnson. - Amiram Eldar, Mar 26 2026

Links

Table of n, a(n) for n=1..99.

Vangalur S. Alagar, The Distribution of the Distance between Random Points, Journal of Applied Probability, Vol. 13, No. 3 (1976), pp. 558-566. See p. 565, eq. (18).

Allan J. Gottlieb (ed.), Puzzle Corner, Technology Review, MIT , March/April 1976, p. 69.

H. J. Oser, Problem 75-12, An Average Distance, SIAM Review, Vol. 17, No. 3 (1975), p. 566; Solution, by J. D. Murchland and D. J. Daley, ibid., Vol. 18, No. 3 (1976), pp. 497-501.

Kenneth J. Travers and Kenneth G. Gray, The Monte Carlo Method: A Fresh Approach to Teaching Probabilistic Concepts, The Mathematics Teacher, Vol. 74, No. 5 (1981), pp. 327-334. See p. 332.

Formula

Equals (116 - 8*sqrt(2) - 20*sqrt(5) + 140*log(1/2 + sqrt(5)/2) - 40*log(1 + sqrt(2)) + 80*log(2 + sqrt(5)) + log(32) + 10*log(-1 + sqrt(5)) - 15*log(123 + 55*sqrt(5))) / 120.

Example

1.0881382498612540260133800398822187615937033186594483235371...

Mathematica

d[x1_, y1_, x2_, y2_] := Sqrt[ (x1-x2)^2+(y1-y2)^2 ]
Integrate[ d[x1, y1, x2, y2], {x1, -1, 0}, {x2, 0, 1}, {y1, 0, 1}, {y2, 0, 1} ]
(* Alternative: *)
RealDigits[(116 - 8 Sqrt[2] - 20 Sqrt[5] + 140 ArcCsch[2] - 40 ArcSinh[1] + 80 ArcSinh[2] + Log[32] + 10 Log[-1 + Sqrt[5]] - 15 Log[123 + 55 Sqrt[5]]) / 120, 10, 120][[1]]

Prog

(PARI) (116 - 8*sqrt(2) - 20*sqrt(5) + 140*log(1/2 + sqrt(5)/2) - 40*log(1 + sqrt(2)) + 80*log(2 + sqrt(5)) + log(32) + 10*log(-1 + sqrt(5)) - 15*log(123 + 55*sqrt(5))) / 120 \\ Amiram Eldar, Mar 26 2026

Crossrefs

Cf. A348668, A394602.

Sequence in context: A388703 A230153 A091648 * A021923 A296496 A065465

Adjacent sequences: A135704 A135705 A135706 * A135708 A135709 A135710

Keyword

nonn,cons

Author

Richard C. Schroeppel and Michael Kleber, Mar 05 2008

Status

approved