A195723 -id:A195723 - cp's OEIS frontend

A137218 Decimal expansion of the argument of -1 + 2*i.

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

Offset: 1

Matt Rieckman (mjr162006(AT)yahoo.com), Mar 06 2008

Gives closed forms for many arctangent values:
arctan(2) = Pi - a, arctan(1/2) = a - Pi/2,
arctan(3) = a - Pi/4, arctan(1/3) = 3*Pi/4 - a,
arctan(7) = 7*Pi/4 - 2*a, arctan(1/7) = 2*a - 5*Pi/4,
arctan(4/3) = 2*a - Pi and arctan(3/4) = 3*Pi/2 - 2*a.
Dihedral angle in the dodecahedron (radians). - R. J. Mathar, Mar 24 2012
Larger interior angle (in radians) of a golden rhombus; A105199 is the smaller interior angle. - Eric W. Weisstein, Dec 17 2018

			2.0344439357957027354455779231...

Rick L. Shepherd, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Golden Rhombus
Wikipedia, Dodecahedron
Index entries for transcendental numbers.

Platonic solids' dihedral angles: A137914 (tetrahedron), A156546 (octahedron), A019669 (cube), A236367 (icosahedron). - Stanislav Sykora, Jan 23 2014
Cf. A242723 (same in degrees).
Cf. A105199 (smaller interior angle of the golden rhombus).

RealDigits[Pi - ArcTan[2], 10, 120][[1]] (* Harvey P. Dale, Aug 08 2014 *)

default(realprecision, 120);
acos(-1/sqrt(5)) \\ or
arg(-1+2*I) \\ Rick L. Shepherd, Jan 26 2014

Equals Pi - arctan(2) = A000796 - A105199 = 2*A195723.

Corrected a typo in the sequence Matt Rieckman (mjr162006(AT)yahoo.com), Feb 05 2010

More terms from Rick L. Shepherd, Jan 26 2014

A300074 Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 01 2018

This is the reciprocal of A182007, and one half of A121570.
This is the ratio of the radius r of the circumscribing circle of a regular pentagon and its side length s: r/s = 1/(2*sin(Pi/5)).
A quartic number of denominator 5 and minimal polynomial 5x^4 - 5x^2 + 1. - Charles R Greathouse IV, Mar 04 2018
Appears at Schur decomposition of A=[1 2; 2 3]. - Donghwi Park, Jun 20 2018

			r/s = 0.850650808352039932181540497063011072240401403764816881836740242377...
2*r/s = A121570.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Wikipedia, Schur decomposition.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A090773, A121570, A182007, A195693, A195723.

RealDigits[1/(2 Sin[Pi/5]), 10, 111][[1]] (* Robert G. Wilson v, Jul 15 2018 *)

1/(2*sin(Pi/5)) \\ Charles R Greathouse IV, Mar 04 2018

sqrt((5+sqrt(5))/10) \\ Charles R Greathouse IV, Mar 04 2018

r/s = 1/A182007 = A121570/2 = (2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = (1 + sqrt(5))/2 = A001622.
From Amiram Eldar, Feb 08 2022: (Start)
Equals cos(arccot(phi)) = cos(arctan(1/phi)) = cos(A195693).
Equals sin(arctan(phi)) = sin(arccot(1/phi)) = sin(A195723). (End)
Equals Product_{k>=1} (1 + (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 2024

A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 04 2011

A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).
Largest root of x^4 - 5x^2 + 5. - Charles R Greathouse IV, May 07 2011
This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012
Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014
The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020

			1.902113032590307144232878666758764286811397268251...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Michael Penn, On the fifth root of the identity matrix, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Golden Rectangle.
Eric Weisstein's World of Mathematics, Pentagram.
Index entries for algebraic numbers, degree 4.

Cf. A001622 (decimal expansion of the golden ratio), A019881.
Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
Cf. A090771, A195693, A195723, A296184.

SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018

r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130][[1]]
RealDigits[Sqrt[GoldenRatio+2],10,130][[1]] (* Harvey P. Dale, Oct 27 2023 *)

PARI
```
sqrt((5+sqrt(5))/2)
```

Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020
Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021
Equals i^(1/5) + i^(-1/5). - Gary W. Adamson, Jul 08 2022
Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - Wolfdieter Lang, Aug 28 2022
Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024

A195693 Decimal expansion of arctan(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 22 2011

Radian measure of half the smaller angle in the golden rhombus. - Eric W. Weisstein, Dec 11 2018

The angle between the diagonal and the longer side of a golden rectangle. - Amiram Eldar, May 18 2021

			arctan(1/phi) = 0.5535743588970452515085327300892685200... .
tan(0.5535743588970452515085327300...) = 1/(golden ratio).
cot(0.5535743588970452515085327300...) = (golden ratio).

Paul S. Bruckman, Problem H-549, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 91; Resurrection, Solution to Problem H-549 by Charles K. Cook, ibid., Vol. 38, No. 2 (2000), pp. 191-192.
Hei-Chi Chan, Machin-type formulas expressing Pi in terms of phi, The Fibonacci Quarterly, Vol. 46/47, No. 1 (2008/2009), pp. 32-37.
Verner E. Hoggatt, Jr. and I. D. Bruggles, A Primer on the Fibonacci Sequence, Part V, The Fibonacci Quarterly, Vol. 2, No. 1 (1964), pp. 59-65.
Eric Weisstein's World of Mathematics, Golden Rhombus.
Index entries for transcendental numbers

Cf. A000032, A000045, A001622, A005248, A175288, A195692, A195694, A195723.

(See also A195692.)
RealDigits[ArcCot[GoldenRatio], 10, 100][[1]] (* or *) RealDigits[(Pi - ArcTan[4/3])/4, 10, 100][[1]] (* Eric W. Weisstein, Dec 11 2018 *)

PARI
```
atan(2)/2 \\ Michel Marcus, Feb 05 2022
```

Equals Pi/2 - A195723. - Amiram Eldar, May 18 2021
Equals arctan(2)/2. - Christoph B. Kassir, Dec 04 2021
From Amiram Eldar, Jan 11 2022: (Start)
Equals arccot(phi).
Equals (Pi - arctan(phi^5))/3.
Equals (Pi - arctan(4/3))/4.
Equals Sum_{k>=1} ((-1)^(k+1) * arctan(1/Fibonacci(2*k))) (Bruckman, 1999). (End)
Equals Sum_{k>=1} arctan(1/Lucas(2*k)) (Hoggatt and Bruggles, 1964). - Amiram Eldar, Feb 05 2022

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

cp's OEIS Frontend

A137218 Decimal expansion of the argument of -1 + 2*i.

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A300074 Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

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A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

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A195693 Decimal expansion of arctan(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

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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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