A073012 Decimal expansion of Robbins constant.

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

Offset

0,1

Comments

The average distance between two points chosen at random inside a unit cube.

This constant was named after the American mathematician David Peter Robbins (1942 - 2003). - Amiram Eldar, Aug 25 2020

Conjecture: this constant is transcendental by Baker's theorem. - Charles R Greathouse IV, Oct 01 2025

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 479.

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 693.

Francois Le Lionnais, Les nombres remarquables, Paris: Hermann, 1983. See p. 30.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Simon Plouffe, The Robbins constant, in Miscellaneous Mathematical Constants, p. 173.

David P. Robbins, Problem E2629, The American Mathematical Monthly, Vol. 84, No. 1 (1977), p. 57, Theodore S. Bolis, Solution to problem E2629: Average distance between two points in a box, also solved by the proposer and by Günter Bach and Frank Piefke, ibid., Vol. 85, No. 4 (1978), pp. 277-278.

Eric Weisstein's World of Mathematics, Cube Line Picking.

Eric Weisstein's World of Mathematics, Hypercube Line Picking.

Eric Weisstein's World of Mathematics, Robbins Constant.

Wikipedia, Robbins constant.

Formula

Equals 4/105 + (17/105) * sqrt(2) - (2/35) * sqrt(3) + (1/5) * log(1+sqrt(2)) + (2/5) * log(2+sqrt(3)) - (1/15) * Pi. - Eric W. Weisstein, Mar 02 2005

Equals (A392512 + A392513 + A392514/3)/5. - Amiram Eldar, Mar 21 2026

Example

0.66170718226717623515583113324841358174640013579095...

Mathematica

RealDigits[ N[4/105 + 17/105*Sqrt[2] - 2/35*Sqrt[3] + 1/5*Log[1 + Sqrt[2]] + 2/5*Log[2 + Sqrt[3]] - 1/15*Pi, 110]] [[1]]

Prog

(PARI) (4 + 17*sqrt(2) - 6*sqrt(3) + 21*log(1 + sqrt(2)) + 42*log(2 + sqrt(3)) - 7*Pi)/105 \\ G. C. Greubel, Jan 11 2017

Crossrefs

Cf. A000796, A091505, A392512, A392513, A392514.

Sequence in context: A153605 A247447 A112302 * A102522 A201672 A200299

Adjacent sequences: A073009 A073010 A073011 * A073013 A073014 A073015

Keyword

cons,nonn,nice

Author

Robert G. Wilson v, Aug 03 2002

Status

approved