A091505 -id:A091505 - cp's OEIS frontend

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

Robert G. Wilson v, Aug 03 2002

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

			0.66170718226717623515583113324841358174640013579095...

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.

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.

Cf. A000796, A091505.

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]]

(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

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

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.

Original entry on oeis.org

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

Amiram Eldar, Oct 29 2021

			0.70980150661400782746374731464451797194994085344545...

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

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.

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

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

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

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

A348681 Decimal expansion of the average length of a chord in a unit square defined by the intersection of the perimeter with a straight line passing through 2 points uniformly and independently chosen at random in the interior of the square.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 29 2021

			1.04281086632944135666196471321448794982806313555801...

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

Rodney Coleman, Random paths through convex bodies, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; alternative link; author's link.

Cf. A091505, A247674, A348680, A348682, A348683.

RealDigits[(2/3) * (Log[1 + Sqrt[2]] + (2 + Sqrt[2])/5), 10, 100][[1]]

(2/3) * (log(1 + sqrt(2)) + (2 + sqrt(2))/5) \\ Michel Marcus, Oct 29 2021

Equals (2/3) * (log(1 + sqrt(2)) + (2 + sqrt(2))/5).

A254140 Decimal expansion of the average reciprocal distance between two points chosen at random in a unit square.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 26 2015

			2.97320959824737870252818566763957179801974547921061 ...

D. H. Bailey, J. M. Borwein, R. E. Crandall, Advances in the theory of box integrals Math. Comp. 79 (2010), 1839-1866, p. 24.
Eric Weisstein's MathWorld, Square Line Picking

Cf. A091505.

RealDigits[4/3 - 4*Sqrt[2]/3 + 4*Log[1 + Sqrt[2]], 10, 105] // First

Integral over a unit square of 1/sqrt((r1-q1)^2 + (r2-q2)^2) dr1 dr2 dq1 dq2 = 4/3 - 4*sqrt(2)/3 + 4*log(1 + sqrt(2)).

A091506 Decimal expansion of (2 + sqrt(2) + 5*arcsinh(1))/9.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 16 2004

Average distance between two points chosen at random on two different edges of a unit square.

			0.86900905527453446388497059434540662485671927963168056966035...

Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery. Natick: A K Peters (2004): p. 66, Example 57(a).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics
Eric Weisstein's World of Mathematics, Square Line Picking
Index entries for transcendental numbers

Cf. A091505.

SetDefaultRealField(RealField(100)); (2 + Sqrt(2) + 5*Argsinh(1))/9; // G. C. Greubel, Aug 17 2018

RealDigits[(2 + Sqrt[2] + 5ArcSinh[1])/9, 10, 120][[1]] (* Harvey P. Dale, May 22 2013 *)

(2 + sqrt(2) + 5*asinh(1))/9 \\ G. C. Greubel, Aug 17 2018

Also equals (2 + sqrt(2) + 5*log(1 + sqrt(2)))/9. - Jean-François Alcover, Feb 14 2014

James D. Klein proved that this constant is equal to 2/3*int(int(sqrt(x^2 + y^2), x = 0..1), y = 0..1) + 1/3*int(int(sqrt(1 + (y - u)^2), u = 0..1), y = 0..1). - John M. Campbell, Apr 02 2016

Broken link fixed by John M. Campbell, Apr 02 2016

A242071 Decimal expansion of 'beta', a constant appearing in the random links Traveling Salesman Problem.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 14 2014

			2.041548186412132418045490158381455866340250252564691915512131281...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.5 Traveling Salesman constants, p. 499.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 60.

Cf. A073008, A091505, A240717.

y[x_] := -2 - ProductLog[-1, E^(-2-x)*(2 - 2*E^x + x)]; beta = (1/2)*NIntegrate[y[x], {x, 0, Infinity}, WorkingPrecision -> 102]; beta // RealDigits // First

beta = integral_{x>0} y(x) dx, where y(x) = -2 - W_(-1) (e^(-2-x) *(2-2*e^x+x)), W_k(z) being the k-th order Lambert W function (also known as ProductLog). y(x) is implicitly defined by the equation (1+x/2)*exp(-x)+(1+y(x)/2)*exp(-y(x)) = 1.

A254149 Decimal expansion of the average reciprocal length of a line segment picked at random in a unit 4-cube.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 26 2015

			1.481432636521064749748769140727658302570952634154861...

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (2010), 1839-1866. See p. 25.
Eric Weisstein's World of Mathematics, Hypercube Line Picking
Eric Weisstein World of Mathematics, Inverse Tangent Integral

Cf. A073012, A091505, A103983, A242588, A254140.

Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); Delta4[-1]=-152/315 - 8*Pi/15 - 16/5*Log[2] + 2/5*Log[3] + 68/105*Sqrt[2] - 16/35*Sqrt[3] + 4/5*Log[1 + Sqrt[2]] + 32/5*Log[1 + Sqrt[3]] - 8/3*Catalan + 8*Ti2[3 - 2 Sqrt[2]] - 8/5*Sqrt[2]*ArcTan[Sqrt[2]/4] // Re; RealDigits[Delta4[-1], 10, 103] // First

from mpmath import *
mp.dps=104
x=3 - 2*sqrt(2)
Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
C=-152/315 - 8*pi/15 - 16/5*log(2) + 2/5*log(3) + 68/105*sqrt(2) - 16/35*sqrt(3) + 4/5*log(1 + sqrt(2)) + 32/5*log(1 + sqrt(3)) - 8/3*catalan + 8*Ti2x - 8/5*sqrt(2)*atan(sqrt(2)/4)
print([int(n) for n in list(str(C.real).replace('.','')[:-1])]) # Indranil Ghosh, Jul 03 2017

Delta_4(-1) = Integral over a unit 4-cube of 1/sqrt((r1-q1)^2+(r2-q2)^2+(r3-q3)^2+(r4-q4)^2) dr dq.

Delta_4(-1) = -152/315 - 8*Pi/15 - 16/5*log(2) + 2/5*log(3) + 68/105*sqrt(2) - 16/35*sqrt(3) + 4/5*log(1 + sqrt(2)) + 32/5*log(1 + sqrt(3)) - 8/3*Catalan + 8*Ti2(3 - 2*sqrt(2)) - 8/5*sqrt(2)*arctan(sqrt(2)/4), where Ti2 is Lewin's arctan integral.

A348669 Decimal expansion of 2sqrt(2)log(1 + sqrt(2))/(3*Pi).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 29 2021

The average length of a random line segment in a unit square defined as follows. A line that is making a random angle with a given edge of the square is chosen, and a random distance of this line from a given vertex of this edge is chosen uniformly between 0 and the distance to the opposite vertex in the square. The segment is then being chosen by picking at random two points between the two intersection points of the line with the perimeter of the square.

			0.26450500700786984551577520129722526936340009096805...

Eric Rosenberg, The expected length of a random line segment in a rectangle, Operations Research Letters, Vol. 32, No. 2 (2004), pp. 99-102.

Cf. A088367, A091505, A091648, A244920.

evalf(sqrt(8/9)*arcsinh(1)/Pi, 120);  # Alois P. Heinz, Oct 29 2021

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

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

cp's OEIS Frontend

A073012 Decimal expansion of Robbins constant.

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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.

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A348681 Decimal expansion of the average length of a chord in a unit square defined by the intersection of the perimeter with a straight line passing through 2 points uniformly and independently chosen at random in the interior of the square.

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A254140 Decimal expansion of the average reciprocal distance between two points chosen at random in a unit square.

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A091506 Decimal expansion of (2 + sqrt(2) + 5*arcsinh(1))/9.

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A242071 Decimal expansion of 'beta', a constant appearing in the random links Traveling Salesman Problem.

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A254149 Decimal expansion of the average reciprocal length of a line segment picked at random in a unit 4-cube.

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A348669 Decimal expansion of 2sqrt(2)log(1 + sqrt(2))/(3*Pi).