A195695 -id:A195695 - cp's OEIS frontend

A137914 Decimal expansion of arccos(1/3).

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

Offset: 1

Rick L. Shepherd, Feb 22 2008

Dihedral angle in radians of regular tetrahedron.
Arccos(1/3) is the central angle of a cube, made by the center and two neighboring vertices. - Clark Kimberling, Feb 10 2009
Also the complementary tetrahedral angle, Pi-A156546, and therefore related to the magic angle (Pi-2*A195696). - Stanislav Sykora, Jan 23 2014
Polar angle (or apex angle) of the cone that subtends exactly one third of the full solid angle. - Stanislav Sykora, Feb 20 2014
Also the acute angle in the rhombi and isosceles trapezoids in the trapezo-rhombic dodecahedron. - Eric W. Weisstein, Jan 09 2019
Also the angle between the tangent lines to the curves y = sin(x) at y = cos(x) at the points of intersection. - David Radcliffe, Jan 17 2023

			1.2309594173407746821349291782479873757103400093550948390555483336639923144...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 58.
Eric Weisstein's World of Mathematics, Tetrahedron.
Eric Weisstein's World of Mathematics, Dihedral Angle.
Eric Weisstein's World of Mathematics, Trapezo-Rhombic Dodecahedron.
Index entries for transcendental numbers

Cf. A137915 (same in degrees), A019670, A195695, A195696, A238238, Platonic solids dihedral angles: A156546 (octahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron).

SetDefaultRealField(RealField(100)); Arccos(1/3); // G. C. Greubel, Aug 20 2018

RealDigits[ArcCos[1/3], 10, 120][[1]] (* Harvey P. Dale, Jul 06 2018 *)
RealDigits[ArcSec[3], 10, 120][[1]] (* Eric W. Weisstein, Jan 09 2019 *)

PARI
```
acos(1/3)
```

arccos(1/3) = arctan(2*sqrt(2)) = 2*arcsin(sqrt(3)/3) = arcsin(2*sqrt(2)/3).
Equals sqrt(2)*Sum_{k>=0} (-1)^k/(2^k*(2*k+1)). - Davide Rotondo, Jun 07 2025
Equals 2*A195695. - Hugo Pfoertner, Jun 07 2025

A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

Angle (in radians) between an edge and (the normal of) a face of the regular tetrahedron. - R. J. Mathar, Feb 23 2012
Also known as magic angle; root of P_2(cos(theta)), with P_2(x) being second-order Legendre polynomial. - Stanislav Sykora, May 25 2012
From Stanislav Sykora, Nov 14 2013: (Start)
Also the angle between the body diagonal of a cube and an incident edge, and therefore the polar angle of the cone circumscribed to a cube from one of its vertices.
Also half of the tetrahedral angle (A156546).
In nuclear magnetic resonance, the angle, with respect to the direction of the main magnetic field, under which a solid sample needs to be spun in order to average to zero unwanted dipole-dipole spin interactions (the magic angle spinning, or MAS, technique). (End)
Also <3_2> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

			0.9553166181245092781638571025157577... (= 54.73561031... degrees).

G. C. Greubel, Table of n, a(n) for n = 0..10000
John H. Conway, Charles Radin, and Lorenzo Sadun, On Angles Whose Squared Trigonometric Functions are Rational, arXiv:math-ph/9812019, 1998. See also Discr. Computat. Geom. (1999) Vol. 22, 321-332.
H. B. Dwight, Tables of Integrals and other Mathematical Data. 507.13, 507.22 in Inverse trigonometric functions. New York: Macmillan Publishing, p. 120, 1961.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022; p. 53.
C. O. Horgan and J. G. Murphy, On an angle with magical properties, Notices Amer. Math. Soc., 69:1 (2022), 22-25.
G. Jacob Martens, Rational right triangles and the Congruent Number Problem, arXiv:2112.09553 [math.GM], 2021, see section 2.6.2 The Trinity vectors, the magic angle, equation (34).
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Eric Weisstein's World of Mathematics, Dehn Invariant.
Wikipedia, Tetrahedron
Wikipedia, Magic angle
Index entries for transcendental numbers

Cf. A156546, A195695, A197739, A210974 (in degrees), A243445.

[Arccos(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

RealDigits[ArcTan[Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Dec 13 2014 *)

atan(sqrt(2)) \\ G. C. Greubel, Jul 05 2017

Equals i*log(sqrt(1/3) - i*sqrt(2/3)). - Andrea Pinos, Nov 03 2023

Equals A156546/2 = 2*A197739. - Hugo Pfoertner, Nov 06 2024

A195698 Decimal expansion of arccos(-sqrt(1/3)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

Dihedral angle between mixed type faces of the truncated octahedron. - R. J. Mathar, Mar 24 2012

			arccos(-1/sqrt(3)) = 2.1862760354...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Truncated Octahedron.
Index entries for transcendental numbers

Cf. A195695.

[Arccos(-1/Sqrt(3))]; // G. C. Greubel, Nov 18 2017

(See A195695.)
RealDigits[ArcCos[-1/Sqrt[3]], 10, 50][[1]] (* G. C. Greubel, Nov 18 2017 *)

acos(-1/sqrt(3)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(2/3)) = Pi - arctan(sqrt(2)). - Amiram Eldar, Jul 08 2023

A118417 a(n) = (2n + 1) 2^(n + 1).

Original entry on oeis.org

2, 12, 40, 112, 288, 704, 1664, 3840, 8704, 19456, 43008, 94208, 204800, 442368, 950272, 2031616, 4325376, 9175040, 19398656, 40894464, 85983232, 180355072, 377487360, 788529152, 1644167168, 3422552064, 7113539584, 14763950080, 30601641984, 63350767616

Offset: 0

Reinhard Zumkeller, Apr 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A014480, A118416.

Cf. A002193, A196525, A195695.

[(2*n+1)*2^(n+1): n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

CoefficientList[Series[2 (1 - 3 x^2 + 2 x^3)/((1 - x)^2 (1 - 2 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 02 2016 *)
Table[(2n+1)2^(n+1),{n,0,30}] (* or *) LinearRecurrence[{4,-4},{2,12},30] (* Harvey P. Dale, Oct 25 2021 *)

a(n) = A118416(n+1,n) = 2*A014480(n).
G.f.: 2*(1-3*x^2+2*x^3)/((1-x)^2*(1-2*x)^2). - Vincenzo Librandi, Sep 02 2016
Sum_{n>=0} 1/a(n) = A196525. - Fred Daniel Kline, May 24 2019
Sum_{n>=0} (-1)^n/a(n) = arctan(1/sqrt(2))/sqrt(2) = A195695 / A002193. - Amiram Eldar, Oct 01 2022

A195701 Decimal expansion of arctan(sqrt(2/3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

			arctan(sqrt(2/3)) = 0.68471920300...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A195695, A195695, A195702.

[Arctan(Sqrt(2/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[2/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195702 *)

atan(sqrt(2/3)) \\ G. C. Greubel, Nov 18 2017

Equals arcsin(sqrt(2/5)) = arccos(sqrt(3/5)). - Amiram Eldar, Jul 10 2023

A195702 Decimal expansion of arccos(-sqrt(2/3)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(-sqrt(2/3)) = 2.5261129449405...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A195701.

[Arccos(-Sqrt(2/3))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[2/3];
N[ArcSin[r], 100]
RealDigits[%]  (* A195696 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A195695 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195702 *)
RealDigits[ArcCos[-Sqrt[(2/3)]],10,120][[1]] (* Harvey P. Dale, Jan 15 2013 *)

acos(-sqrt(2/3)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(1/3)) = Pi - arctan(sqrt(1/2)). - Amiram Eldar, Jul 10 2023

A243445 Decimal expansion of the polar angle of the cone circumscribed to a regular dodecahedron from one of its vertices.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jun 06 2014

The angle is in radians.

			1.20593249868141343750392336173309109440033174266369606513299755...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Dodecahedron (use the point coordinates to derive the formula).

Cf. A001622 (phi), A003881 (octahedron), A195695 (tetrahedron), A195696 (cube), A195723 (isosahedron).

RealDigits[ArcCos[1/(GoldenRatio Sqrt[3])],10,120][[1]] (* Harvey P. Dale, May 17 2016 *)

PARI
```
acos(2/(1+sqrt(5))/sqrt(3))
```

arccos(1/(phi*sqrt(3))), where phi = A001622.

arctan(phi^2), where phi = A001622. - Jon Maiga, Nov 11 2018

A377202 Decimal expansion of Integral_{x=0..oo} exp(-x)*erf(sqrt(x))^2 dx, where erf is the error function.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Oct 19 2024

			0.554126423979571987123078834537902718922949579396...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Erf.
Wikipedia, Error function.

Cf. A000796, A010466, A195695, A377203.

First[RealDigits[Sqrt[8]*ArcCot[Sqrt[2]]/Pi, 10, 100]]

Equals 2*sqrt(2)*arccot(sqrt(2))/Pi = A010466*A195695/A000796 (cf. eq. 37 in Weisstein link).

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A137914 Decimal expansion of arccos(1/3).

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A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

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A195698 Decimal expansion of arccos(-sqrt(1/3)).

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A118417 a(n) = (2*n + 1) * 2^(n + 1).

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A195701 Decimal expansion of arctan(sqrt(2/3)).

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A195702 Decimal expansion of arccos(-sqrt(2/3)).

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A243445 Decimal expansion of the polar angle of the cone circumscribed to a regular dodecahedron from one of its vertices.

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A118417 a(n) = (2n + 1) 2^(n + 1).