A173201 -id:A173201 - cp's OEIS frontend

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

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;

A133731 Decimal expansion of goat tether length to graze half a unit field.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 21 2007

See Ullisch link for a closed form. - Charles R Greathouse IV, Jul 08 2023

			1.1587284730181215178...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.
Ingo Ullisch, A Closed-Form Solution to the Geometric Goat Problem, The Mathematical Intelligencer volume 42 (2020), pp. 12-16.

Michael Johannes Latsky, Table of n, a(n) for n = 1..20000
M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227. Has extensive bibliography.
James Grime and Brady Haran, The Goat Problem, Numberphile video (2023).
Graham Jameson and Nicholas Jameson, Goats and birds, The Mathematical Gazette, Volume 101, Issue 551 (July 2017), pp. 296-300.
Gerd Lamprecht, A133731=cos(A173201/2)*2; 10000 digits [Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010]
Gerd Lamprecht, Iterationsrechner mit Algorithmus [Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010]
Eric Weisstein's World of Mathematics, Goat Problem.

A173201 = x /. FindRoot[(-x*Cos[x] + Sin[x] - Pi/2)/(Sin[x]*x), {x, 1}, WorkingPrecision -> 105]; RealDigits[2*Cos[A173201/2] ][[1]] (* Jean-François Alcover, Oct 31 2012 *)

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

Equals cos(A173201/2)*2. - Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010

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

Original entry on oeis.org

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

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

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A133731 Decimal expansion of goat tether length to graze half a unit field.

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

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