A133731 -id:A133731 - cp's OEIS frontend

A192930 Decimal expansion of Pi*cos(phi) - Pi/2, where phi is the constant defined by A191102.

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

Offset: 0

N. J. A. Sloane, Jul 12 2011

Length of the rope in the "goat outside the fence" version of the grazing goat problem, when the radius of the circular field is assumed to be 1. See Fraser (1982) for details. - Hugo Pfoertner, Apr 05 2020

			0.91508964079634209342198381417593107109296289714973860113292...

G. C. Greubel, Table of n, a(n) for n = 0..10000
M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227.

Cf. A133731, A173201.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)*( Cos(Arccos(6/Pi(R)^2 -1)/3) -1/2); // G. C. Greubel, Feb 06 2019

RealDigits[Pi*(Cos[ArcCos[6/Pi^2 -1]/3] -1/2), 10, 100][[1]] (* G. C. Greubel, Feb 06 2019 *)

Pi*cos(acos(6/Pi^2-1)/3) - Pi/2 \\ Michel Marcus, Sep 19 2017

numerical_approx(pi*(cos(acos(6/pi^2 -1)/3) - 1/2), digits=100) # G. C. Greubel, Feb 06 2019

Equals (z + 1/z - 1)*Pi/2 where x = 6/Pi^2 - 1 and z = (x - sqrt(x^2 - 1))^(1/3). - Peter Luschny, Apr 05 2020

A352453 Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 16 2022

The solution to a problem in Jones (1932): "At each corner of a garden, surrounded by a wall n yards square, a goat is tied with a rope n yards long. Find the area of the part of the garden common to the four goats." (When the square is taken to be of unit size, the common area is this constant.)
The perimeter of the shape formed by the intersection is 2*Pi/3 (A019693).
The solution to the three-dimensional version of this problem is A352454.

			0.31514674362772045262676811958729526112291787931465...

Donald L. Chambers, Problem 3684, School Science and Mathematics, Vol. 77, No. 5 (1977), p. 443; Solution by J. Philip Smith, ibid., Vol. 78, No. 4 (1978), pp. 354-355.
Amiram Eldar, Illustration.
Samuel Isaac Jones, Mathematical Nuts: For Lovers of Mathematics, 1932, Problems 9 and 10, pp. 86, 301-302.
Missouri State University, Problem #8, Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres), Advanced Problem Archive,; Solution to Problem #8, by Raymond Roan.
Bruce Shawyer, Problem 6, APICS 1999 Mathematics Competition, The Academy Corner, Crux Mathematicorum, Vol. 25, No. 8, 1999, p. 453; Solutions by Richard Tod and Catherine Shevlin, Vol. 26, No. 4, 2000, pp. 193-194.
Charles W. Trigg, Problem 686, Crux Mathematicorum, Vol. 7, No. 9, 1981, p. 275; Solution by Jordan Dou, Vol. 8, No. 9, 1982, p. 294.

Cf. A002194, A019670, A019693.

Cf. A075838, A133731, A192930, A352454.

RealDigits[1 + Pi/3 - Sqrt[3], 10, 100][[1]]

Equals 1 + Pi/3 - sqrt(3) = 1 + A019670 - A002194.

A173201 Fixed point of the iteration x |--> x - (sin(x) - cos(x)x - Pi/2)/(sin(x)x).

Original entry on oeis.org

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

Offset: 1

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010

Decimal expansion of psi, the unique solution on (0,Pi) of sin(psi) - psi*cos(psi) = Pi/2, an auxiliary constant used in the Hall-Tenenbaum inequality applied to real multiplicative functions. - Jean-François Alcover, Sep 05 2014

			A133731 = cos(1.9056957293.../2)*2.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.
M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227.
Gerd Lamprecht, Iterationsrechner mit Algorithmus
Gerd Lamprecht, 10000 digits
Gerd Lamprecht, Zahlenfolgen (sequence)

Cf. A072112, A133731.

psi = x /. FindRoot[Sin[x] - x*Cos[x] == Pi/2, {x, 2}, WorkingPrecision -> 102]; RealDigits[psi] // First (* Jean-François Alcover, Sep 05 2014 *)

solve(x=1,2,sin(x)-x*cos(x)-Pi/2) \\ Charles R Greathouse IV, Mar 03 2021

x := x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).

Equals 2*arccos(A133731/2).

A075838 Decimal expansion of the solution to the donkey problem.

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Oct 17 2002

			0.95284786465494194744133321858048335174752156080640...

Dr. Math, Grazing Animals.
Marshall Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227. [_N. J. A. Sloane_, Jul 12 2011]

Cf. A133731, A173571, A192930.

RealDigits[x /. FindRoot[4*x*Cos[x]^2 + Pi/2 - 2*x - Sin[2*x] == 0, {x, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* Amiram Eldar, Apr 29 2023 *)

solve(x=0, 1, 4*x*cos(x)^2 + Pi/2 - 2*x - sin(2*x)) \\ Michel Marcus, Sep 19 2017

x: 4x*cos^2(x) + (1/2)Pi - 2x - sin(2x) = 0.

A336198 Decimal expansion of the radius of a sphere centered on the surface of a unit-radius sphere and dividing it into two parts of equal volume.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 11 2020

The solution to the grazing goat problem in three dimensions.

			1.228544863735220903448994497685293465644191645518602...

Marshall Fraser, The Grazing Goat in n Dimensions, The Two-Year College Mathematics Journal, Vol. 15, No. 2 (1984), pp. 126-134.
Mark D. Meyerson, Return of the Grazing Goat in n Dimensions, The College Mathematics Journal, Vol. 15, No. 5 (1984), pp. 430-432.
Eric Weisstein's World of Mathematics, Goat Problem.
Wikipedia, Goat problem.

Cf. A019699, A133731.

RealDigits[x /. Solve[3*x^4 - 8*x^3 + 8 == 0 && x > 0, {x}, Reals][[1]], 10, 100][[1]]

The smaller of the 2 real roots of the equation 3*x^4 - 8*x^3 + 8 = 0.

A173571 Decimal expansion of constant related to Goat Problem, Donkey Problem, Tenenbaum and A173201.

Original entry on oeis.org

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

Offset: 0

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 22 2010

			0.9528478646... = A075838 =(0.9444433782055...+PI/2-asin(0.9444433782055...))/(A133731^2);

M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227. Has extensive bibliography. [From _N. J. A. Sloane_, Jul 12 2011]
Gerd Lamprecht, Iterationsrechner Beispiel 3

x = sqrt(1-A072112^2) = sqrt(1-(1-A133731^2/2)^2); A075838 =(x+PI/2-asin(x))/(A133731^2); with A072112=1-A133731^2/2; A133731=cos(A173201/2)*2;

cp's OEIS Frontend

A192930 Decimal expansion of Pi*cos(phi) - Pi/2, where phi is the constant defined by A191102.

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A352453 Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.

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A173201 Fixed point of the iteration x |--> x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).

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A075838 Decimal expansion of the solution to the donkey problem.

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A336198 Decimal expansion of the radius of a sphere centered on the surface of a unit-radius sphere and dividing it into two parts of equal volume.

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A173571 Decimal expansion of constant related to Goat Problem, Donkey Problem, Tenenbaum and A173201.

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A173201 Fixed point of the iteration x |--> x - (sin(x) - cos(x)x - Pi/2)/(sin(x)x).