A195692 -id:A195692 - cp's OEIS frontend

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

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

A175288 Decimal expansion of the minimal positive constant x satisfying (cos(x))^2 = sin(x).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 23 2010, Mar 29 2010

This is the angle (in radians) at which the modified loop curve x^4=x^2*y-y^2 returns to the origin. Writing the curve in (r,phi) circular coordinates, r = sin(phi) * (cos^2(phi)-sin(phi)) /cos^4(phi), the two values of r=0 are phi=0 and the value of phi defined here. The equivalent angle of the Bow curve is Pi/4.

Also the minimum positive solution to tan(x) = cos(x). - Franklin T. Adams-Watters, Jun 17 2014

			x = 0.66623943.. = 38.1727076... degrees.

Eric Weisstein's World of Mathematics, Bow.

Cf. A019669, A094214, A001622, A195692, A352151.

r = 1/GoldenRatio;
N[ArcSin[r], 100]
RealDigits[%]  (* A175288 *)
RealDigits[x/.FindRoot[Cos[x]^2==Sin[x],{x,.6}, WorkingPrecision->120]] [[1]] (* Harvey P. Dale, Nov 08 2011 *)
RealDigits[ ArcCos[ Sqrt[ (Sqrt[5] - 1)/2]], 10, 105] // First (* Jean-François Alcover, Feb 19 2013 *)

x = arcsin(A094214). cos(x)^2 = sin(x) = 0.618033988... = A094214.
From Amiram Eldar, Feb 07 2022: (Start)
Equals Pi/2 - A195692.
Equals arccos(1/sqrt(phi)).
Equals arctan(1/sqrt(phi)) = arccot(sqrt(phi)). (End)
Root of the equation cos(x) = tan(x). - Vaclav Kotesovec, Mar 06 2022

Disambiguated the curve here from the Mathworld bow curve - R. J. Mathar, Mar 29 2010

A195694 Decimal expansion of arccos(-1/r), where r = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 22 2011

Let c1 and c2 be the constants at A195692 and A195694; then c1+c2 = Pi.

			arccos(-1/r) = 2.2370357592874118743353265876175441...

Index entries for transcendental numbers

Cf. A001622, A175288, A195692, A195693.

(See A195692.)
RealDigits[ArcCos[-1/GoldenRatio],10,120][[1]] (* Harvey P. Dale, Jan 15 2012 *)

acos(-2/(sqrt(5)+1)) \\ Charles R Greathouse IV, Nov 21 2024

A342571 Decimal expansion of the surface area of a golden ellipsoid with semi-axes lengths 1, 1 and phi (A001622).

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Mar 27 2021

			17.9807974341047734215245495904396388204265935060073...

Kenneth Brecher, The "PhiTOP": A Golden Ellipsoid, Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, 2015, pp. 371-374.
Kenneth Brecher and Rod Cross, Physics of the PhiTOP, The Physics Teacher, Vol. 57, No. 2 (2019), pp. 74-75.
Eric Weisstein's World of Mathematics, Ellipsoid.
Wikipedia, Ellipsoid.

Cf. A001622, A195692, A309282.

RealDigits[SurfaceArea[Ellipsoid[{0,0,0},{1,1,GoldenRatio}]], 10, 100][[1]]
(* requires Mathematica 12+, or *)
RealDigits[2*Pi*(1 + GoldenRatio/Sinc[ArcCos[1/GoldenRatio]]), 10, 100][[1]]

Equals 2*Pi*(1 + phi*c/sin(c)), where c = arccos(1/phi) (A195692).

Equals 2*Pi*(1 + sqrt(2+sqrt(5))*arcsec(phi)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A175288 Decimal expansion of the minimal positive constant x satisfying (cos(x))^2 = sin(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A195694 Decimal expansion of arccos(-1/r), where r = (1+sqrt(5))/2 (the golden ratio).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A342571 Decimal expansion of the surface area of a golden ellipsoid with semi-axes lengths 1, 1 and phi (A001622).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula