A244921 -id:A244921 - cp's OEIS frontend

A103712 Decimal expansion of the expected distance from a randomly selected point in the unit square to its center: (sqrt(2) + log(1 + sqrt(2)))/6.

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

Offset: 0

Sylvester Reese and Jonathan Sondow, Feb 13 2005

Is it a coincidence that this constant is equal to 1/6 of the universal parabolic constant A103710? (Reese, 2004; Finch, 2012)
exp(d(2)) - exp(d(2))/Pi = 0.9994179247351742... ~ 1 - 1/1718. - Gerald McGarvey, Feb 21 2005
Take a point on a line of irrational slope and a line segment of a given length centered at the point, integrate the distance of a point on the line to the set of lattice points along the line segment, and divide by the length. The limit as the length approaches infinity can be shown by a generalization of the Equidistribution Theorem to give the expected distance of a point in the unit square to its corners, this constant. - Thomas Anton, Jun 19 2021

			0.38259785823210634567238300819824839793297203393976391398832922444...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 8.1.
S. Reese, A universal parabolic constant, 2004, preprint.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Steven R. Finch, Mathematical Constants, Errata and Addenda, 2012, section 8.1.
Sylvester Reese, Pohle Colloquium Video Lecture: The universal parabolic constant, February 2005.
Sylvester Reese, Jonathan Sondow and Eric W. Weisstein, MathWorld: Universal Parabolic Constant.
Eric Weisstein's World of Mathematics, Universal Parabolic Constant.
Eric Weisstein's World of Mathematics, Square Line Picking.
Wikipedia, Universal parabolic constant.
Index entries for transcendental numbers.

Equal to (A002193 + A091648)/6 = (A103710)/6 = (A103711)/3.

Cf. A244921.

RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/6, 10, 111][[1]] (* Robert G. Wilson v, Feb 14 2005 *)

fpprec: 100$ ev(bfloat((sqrt(2) + log(1 + sqrt(2)))/6)); /* Martin Ettl, Oct 17 2012 */

(sqrt(2) + log(1 + sqrt(2)))/6 \\ G. C. Greubel, Sep 22 2017

Equals (1/3)*Integral_{x = 0..1} sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019
Equals Integral_{x>=1} arcsinh(x)/x^4 dx. - Amiram Eldar, Jun 26 2021
Equals A244921 / 2. - Amiram Eldar, Jun 04 2023

A247674 Decimal expansion of the integral over the square [0,1]x[0,1] of sqrt(1+(x-y)^2) dx dy.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 22 2014

The average length of chords in a unit square drawn between two points uniformly and independently chosen at random on two opposite sides. - Amiram Eldar, Aug 08 2020

			1.076635732895178008965379750243226282838269703135986...

G. C. Greubel, Table of n, a(n) for n = 1..10000
D. H. Bailey and J. M. Borwein, Highly Parallel, High-Precision Numerical Integration, Lawrence Berkeley National Laboratory (2005), p. 9.
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. A244921.

SetDefaultRealField(RealField(100)); R:= RealField(); (2 + Sqrt(2) + 5*Log(1+Sqrt(2)))/3; // G. C. Greubel, Aug 31 2018

RealDigits[2/3 - Sqrt[2]/3 + ArcSinh[1], 10, 103] // First

default(realprecision, 100); (2 + sqrt(2) + 5*log(1+sqrt(2)))/3 \\ G. C. Greubel, Aug 31 2018

Equals 2/3 - sqrt(2)/3 + arcsinh(1).

Equals 2*A244921 + A247674 = (2 + sqrt(2) + 5*log(1+sqrt(2)))/3.

A247685 Decimal expansion of the integral over the square (0,1)x(0,1) of 1/((x+y)sqrt((1-x)(1-y))) dx dy.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 22 2014

Also hyperbolic volume of the Whitehead link complement and (-2,3,8) pretzel link complement. This is the minimal volume attainable by a two-cusped orientable hyperbolic 3-manifold. - Jeremy Tan, Nov 17 2016

			3.663862376708876060218414059729536443096597497126688537...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Ian Agol, The minimal volume orientable hyperbolic 2-cusped 3-manifolds, Proc. Amer. Math. Soc., Vol. 138, No. 10 (2010), pp. 3723-3732. MR2661571
David H. Bailey and Jonathan M. Borwein, Highly Parallel, High-Precision Numerical Integration, Lawrence Berkeley National Laboratory, 2005, p. 9.
Eric Weisstein's World of Mathematics, Lerch Transcendent.
Wikipedia, Lerch zeta function.

Cf. A006752 (Catalan), A091518, A244921, A247674, A247675, A247677, A247684.

SetDefaultRealField(RealField(100)); R:=RealField(); 4*Catalan(R); // G. C. Greubel, Aug 25 2018

evalf(4*Catalan, 130);  # Alois P. Heinz, Aug 14 2023

Mathematica
```
RealDigits[4*Catalan, 10, 103] // First
```

default(realprecision, 100); 4*Catalan \\ G. C. Greubel, Aug 25 2018

lerchphi(-1, 2, 1/2) \\ Charles R Greathouse IV, Jan 30 2025

sumalt(k=0, (-1)^k/(k+1/2)^2) \\ Charles R Greathouse IV, Jan 30 2025

Equals 4*Catalan.
Equals Integral_{x=0..Pi/2} log((1+cos(x))/(1-cos(x))) dx = Integral_{x=0..Pi/2} log((1+sin(x))/(1-sin(x))) dx. - Amiram Eldar, Apr 07 2022
From Amiram Eldar, Aug 14 2023: (Start)
Equals Phi(-1, 2, 1/2) = Sum_{k>=0} (-1)^k/(k+1/2)^2, where Phi is the Lerch transcendent.
Equals Integral_{x=-Pi/2..Pi/2} x/sin(x) dx. (End)

A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 10 2007

			0.960591956455052959425107951...

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2. [From _R. J. Mathar_, Oct 13 2010]
Eric Weisstein's World of Mathematics, Box Integral.

Cf. A010527, A019691, A065914, A135691.

Analogous constants: A244921 (square), A254979 (4-cube).

evalf( sqrt(3)/4+log(2+sqrt(3))/2-Pi/24);

RealDigits[Sqrt[3]/4 + Log[2+Sqrt[3]]/2 - Pi/24, 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

Equals sqrt(3)/4 + log(2+sqrt(3))/2 - Pi/24 = A010527/2 + A065914/2 - A019691.

Equals 2 * A135691. - Amiram Eldar, Jun 04 2023

Name corrected by Amiram Eldar, Jun 04 2023

A247675 Decimal expansion of the integral over the square [-1,1]x[-1,1] of 1/sqrt(1+x^2+y^2) dx dy.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 22 2014

			3.17343648530607134219175646704553851997648161961999537...

G. C. Greubel, Table of n, a(n) for n = 1..10000
D. H. Bailey, J. M. Borwein, Highly Parallel, High-Precision Numerical Integration p. 9. (2005) Lawrence Berkeley National Laboratory

Cf. A244921, A247674.

SetDefaultRealField(RealField(100)); R:= RealField(); 4*Log(2 + Sqrt(3)) - 2*Pi(R)/3; // G. C. Greubel, Aug 31 2018

RealDigits[4*Log[2 + Sqrt[3]] - 2*Pi/3, 10, 103] // First

default(realprecision, 100); 4*log(2 + sqrt(3)) - 2*Pi/3 \\ G. C. Greubel, Aug 31 2018

Equals 4*log(2 + sqrt(3)) - 2*Pi/3.

A254979 Decimal expansion of the mean Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(1) in Bailey's paper).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 11 2015

Also, decimal expansion of twice the expected distance from a randomly selected point in the unit 4D cube to the center. - Amiram Eldar, Jun 04 2023

			1.12189961871586097735161517556754270920080795643954583...

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2.
Eric Weisstein's World of Mathematics, Inverse Tangent Integral.
Eric Weisstein's World of Mathematics, Polylogarithm.
Eric Weisstein's World of Mathematics, Box Integral.

Cf. A117653, A117654, A244920, A254968.

Analogous constants: A244921 (square), A130590 (cube).

Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[1] = 2/5 - Catalan/10 + (3/10)*Ti2[3 - 2*Sqrt[2]] + Log[3] - (7*Sqrt[2]/10)*ArcTan[1/Sqrt[8]] // Re; RealDigits[B4[1], 10, 105] // First
N[Integrate[1/u^2 - Pi^2*Erf[u]^4/(16*u^6), {u, 0, Infinity}]/Sqrt[Pi], 50] (* Vaclav Kotesovec, Aug 13 2019 *)

from mpmath import *
mp.dps=106
x=3 - 2*sqrt(2)
Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
C = 2/5 - catalan/10 + (3/10)*Ti2x + log(3) - (7*sqrt(2)/10)*atan(1/sqrt(8))
print([int(n) for n in str(C.real).replace('.', '')]) # Indranil Ghosh, Jul 04 2017

Equals B_4(1) = 2/5 - Catalan/10 + (3/10)*Ti_2(3-2*sqrt(2)) + log(3) - (7*sqrt(2)/10) * arctan(1/sqrt(8)), where Ti_2(x) = (i/2)*(polylog(2, -i*x) - polylog(2, i*x)) (Ti_2 is the inverse tangent integral function).

Name corrected by Amiram Eldar, Jun 04 2023

A244922 Decimal expansion of the integral over the square [0,1]x[0,1] of (x^2 + y^2)^(3/2) dx dy.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 08 2014

			0.627180784883514720865482452220363173853609205621177137224832249594762945...

D. H. Bailey, J. M. Borwein, R. E. Crandall, Advances in the theory of box integrals (2010) p. 22.

Cf. A244920, A244921.

RealDigits[7/20*Sqrt[2] + 3/20*Log[1 + Sqrt[2]], 10, 106] // First

7/20*sqrt(2) + 3/20*log(1 + sqrt(2)).

Also equals (7*sqrt(2) + 3*arcsinh(1))/20.

A247677 Decimal expansion of the integral over the square [0,Pi]x[0,Pi] of log(2-cos(x)-cos(y)) dx dy.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 22 2014

			4.6692746625755280622628415121320951291596385520228...

G. C. Greubel, Table of n, a(n) for n = 1..10000
D. H. Bailey, J. M. Borwein, Highly Parallel, High-Precision Numerical Integration p. 9. (2005) Lawrence Berkeley National Laboratory

Cf. A244921, A247674, A247675.

SetDefaultRealField(RealField(100)); R:= RealField(); 4*Pi(R)*Catalan(R) - Pi(R)^2*Log(2); // G. C. Greubel, Aug 31 2018

RealDigits[4*Pi*Catalan - Pi^2*Log[2], 10, 104] // First

4*Pi*Catalan - Pi^2*log(2) \\ Michel Marcus, Sep 22 2014

Equals 4*Pi*Catalan - Pi^2*log(2).

A355183 Decimal expansion of the area of the region that represents the set of points in a unit square that are closer to the center of the square than to the closest edge.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 23 2022

The shape is formed by the intersection of four parabolas. Its perimeter is given in A355184.

			0.21895141649746006506891829894626410475956250050259...

Kiran S. Kedlaya, Bjorn Poonen, and Ravi Vakil, The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary, The Mathematical Association of America, 2002, pp. 108-109.

Joel Atkins, Regular Polygon Targets, Pi Mu Epsilon Journal, Vol. 9, No. 3 (1989), pp. 142-144; entire issue.
Nicholas R. Baeth, Loren Luther, and Rhonda McKee, The Downtown Problem: Variations on a Putnam Problem, Mathematics Magazine, Vol. 90, No. 4 (2017), pp. 243-257.
John Coffey, Q19, Maths Puzzles & Problems, MathStudio, 2011.
Amiram Eldar, Illustration.
Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Fiftieth William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 98, No. 4 (1991), pp. 319-327.
Missouri State University, Problem #5, The Area and Perimeter of a Certain Region, Advanced Problem Archive; Solution to Problem #5, by John Shonder.
Jun-Ping Shi, Problem Set 9.

Cf. A021058, A103712, A244921, A254140, A352453, A355184 (perimeter), A355185 (3D analog).

RealDigits[(4*Sqrt[2] - 5)/3, 10, 100][[1]]

Equals (4*sqrt(2)-5)/3.

A247684 Decimal expansion of the integral over the first quadrant (x>0, y>0) of sqrt(x^2 + xy + y^2)exp(-x-y) dx dy.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 22 2014

			1.8239592165010822685464339276918942784856179183670620888...

G. C. Greubel, Table of n, a(n) for n = 1..10000
D. H. Bailey, J. M. Borwein, Highly Parallel, High-Precision Numerical Integration p. 9. (2005) Lawrence Berkeley National Laboratory

Cf. A244921, A247674, A247675, A247677.

SetDefaultRealField(RealField(100)); (4 + 3*Log(3))/4; // G. C. Greubel, Sep 07 2018

RealDigits[1 + (3/4)*Log[3], 10, 102] // First

default(realprecision, 100); (4 + 3*log(3))/4 \\ G. C. Greubel, Sep 07 2018

Equals 1 + (3/4)*log(3).

cp's OEIS Frontend

A103712 Decimal expansion of the expected distance from a randomly selected point in the unit square to its center: (sqrt(2) + log(1 + sqrt(2)))/6.

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A247674 Decimal expansion of the integral over the square [0,1]x[0,1] of sqrt(1+(x-y)^2) dx dy.

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A247685 Decimal expansion of the integral over the square (0,1)x(0,1) of 1/((x+y)*sqrt((1-x)*(1-y))) dx dy.

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A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

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A247675 Decimal expansion of the integral over the square [-1,1]x[-1,1] of 1/sqrt(1+x^2+y^2) dx dy.

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A254979 Decimal expansion of the mean Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(1) in Bailey's paper).

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A247685 Decimal expansion of the integral over the square (0,1)x(0,1) of 1/((x+y)sqrt((1-x)(1-y))) dx dy.

A247684 Decimal expansion of the integral over the first quadrant (x>0, y>0) of sqrt(x^2 + xy + y^2)exp(-x-y) dx dy.