A019669 -id:A019669 - cp's OEIS frontend

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

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

A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 19 2011

The hyperbolas y^2-x^2=1 and xy=1 meet at (1/c,c) and (-1/c,c), where c=sqrt(golden ratio); see the Mathematica program for a graph; see A189339 for hyperbolas meeting at (c,1/c) and (-c,-1/c).
This number is the eccentricity of an ellipse inscribed in a golden rectangle. - Jean-François Alcover, Sep 03 2015
c/sqrt(-1) is the limit of Pi(a;n)/2 := a^n * sqrt(a - f(a;n))  with f(a;0) = 0, and f(a;n) = sqrt(a + f(a;n-1)) for n >= 1, if one takes a = 1. For a=2 this gives Viète's formula for Pi/2 (see A019669). - Wolfdieter Lang, Jul 06 2018

			0.786151377757423286069558585842958929523122057...

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A001622, A019669, A139339.

N[1/Sqrt[GoldenRatio], 110]
RealDigits[%]
FindRoot[x*Sqrt[1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]
Plot[{Sqrt[1 + x^2], 1/x}, {x, 0, 3}]

sqrt(2/(1+sqrt(5))) \\ Michel Marcus, Sep 03 2015

my(c=1/quadgen(5)); a_vector(len) = digits(sqrtint(floor(c*100^len))); \\ Kevin Ryde, Jul 12 2025

Equals sqrt(1/phi) = sqrt(phi-1), with phi = A001622.
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139339.
Equals tan(arcsin(1/phi)).
Equals sin(arccos(1/phi)).
Equals cos(arcsin(1/phi)).
Equals cot(arccos(1/phi)). (End)

A232247 Decimal expansion of the arctan of 2/Pi.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Nov 21 2013

			0.56691150494100940508289774672261915380648023909268233575775947204589...

Cf. A060294, A232182.

Maple
```
evalf(arctan(2/Pi));
```
Mathematica
```
RealDigits[ArcTan[2/Pi], 10, 90][[1]]
```

atan(2/Pi) \\ Charles R Greathouse IV, Mar 24 2021

Equals A019669 - A232182.

Equals Sum_{k>=0} (-1)^k*(2/Pi)^(1+2*k)/(1+2*k).

A257817 Decimal expansion of the real part of li(i), i being the imaginary unit.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 10 2015

li(x) is the logarithmic integral function, extended to the whole complex plane. The corresponding imaginary part is in A257818.

			0.47200065143956865077760610761412783650733054301836188186838371...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function

Cf. A001620, A019669, A094642, A257818.

evalf(Re(Li(I)),120); # Vaclav Kotesovec, May 10 2015

RealDigits[Re[LogIntegral[I]], 10, 120][[1]] (* Vaclav Kotesovec, May 10 2015 *)

li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
  if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
  return(+Pi*I-eint1(-log(c)))); }
  a=real(li(I))

Equals gamma + log(Pi/2) + Sum_{k>=1}((-1)^k*(Pi/2)^(2*k)/(2*k)!/(2*k)).

Equals Ci(Pi/2), the maximum value of the cosine integral along the real axis. - Stanislav Sykora, Nov 12 2016

A257818 Decimal expansion of the imaginary part of li(i), i being the imaginary unit.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, May 10 2015

li(x) is the logarithmic integral function, extended to the whole complex plane. The corresponding real part is in A257817.

			2.941558494949385099300999980021326772089446035251921590122704439...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function

Cf. A019669, A257817.

evalf(Im(Li(I)), 120); # Vaclav Kotesovec, May 10 2015
evalf(Pi/2*(1+Sum(((-1)^k*(Pi/2)^(2*k)/(2*k+1)!/(2*k+1)), k=0..infinity)), 120); # Vaclav Kotesovec, May 10 2015

RealDigits[Im[LogIntegral[I]], 10, 120][[1]] (* Vaclav Kotesovec, May 10 2015 *)

li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
  if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
  return(+Pi*I-eint1(-log(c))));}
  a=imag(li(I))

Equals (Pi/2)*(1+Sum_{k>=0}((-1)^k*(Pi/2)^(2*k)/(2*k+1)!/(2*k+1))).

A352125 Decimal expansion of Pisqrt(2)sqrt(2 + sqrt(2))/8.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Mar 05 2022

			1.02617215297703088887146778087283197497962...

Jean-François Pabion, Éléments d'Analyse Complexe, licence de Mathématiques, page 111, Ellipses, 1995.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, pp. 235-236.
Index entries for transcendental numbers.

Cf. A000796, A047522, A121601.

Integral_{x=0..oo} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), A019670 (m=6), this sequence (m=8), A094888 (m=10).

First[RealDigits[N[Pi*Sqrt[2]Sqrt[2+Sqrt[2]]/8,95]]]

Pi*sqrt(4 + 2*sqrt(2))/8 \\ Michel Marcus, Mar 07 2022

Equals Integral_{x=0..oo} 1/(1 + x^8) dx.
Equals Pi*csc(Pi/8)/8.
Equals 1/Product_{k>=1} (1 - 1/(8*k)^2). - Amiram Eldar, Mar 12 2022
Equals Product_{k>=2} (1 + (-1)^k/A047522(k)). - Amiram Eldar, Nov 22 2024

A378354 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a (small) triakis octahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 24 2024

The (small) triakis octahedron is the dual polyhedron of the truncated cube.

			2.57174440034566846791285405092806379355115694111...

Wikipedia, Triakis octahedron.

Cf. A378351 (surface area), A378352 (volume), A378353 (inradius), A201488 (midradius).
Cf. A019669 and A195698 (dihedral angles of a truncated cube).
Cf. A377342.

First[RealDigits[ArcCos[-(3 + 8*Sqrt[2])/17], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TriakisOctahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(3 + 8*sqrt(2))/17) = arccos(-(3 + A377342)/17).

A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 22 2011

Every cyclic quadrilateral all of whose angles are greater than arccos((sqrt(5)-1)/2) admits a 3 × 1 grid dissection into three cyclic quadrilaterals [Thm. 2.3 in Choi et al. p. 2]. - Michel Marcus, Aug 13 2019
The base angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
Smallest positive root of the equation sin(x) = cot(x). - Wolfe Padawer, Apr 11 2023

			arccos(1/phi) = 0.904556894302381364127316795661958721...
cos(0.904556894302381364127316795661958721...) = 1/(golden ratio) = 0.618...
sec(0.904556894302381364127316795661958721...) = (golden ratio) = 1.618...

Erica Choi, Dan Ismailescu, Jiho Lee and Joonsoo Lee, Grid dissections of tangential quadrilaterals, arXiv:1908.02251 [math.MG], 2019.
Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
Index entries for transcendental numbers

Cf. A001622, A019669, A139339, A175288, A195693, A195694, A197762.

r = 1/GoldenRatio;
N[ArcCos[r], 100]
RealDigits[%]

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

From Amiram Eldar, Feb 07 2022: (Start)
Equals Pi/2 - A175288.
Equals arcsin(1/sqrt(phi)).
Equals arctan(sqrt(phi)). (End)

Terms replaced with intended terms by Rick L. Shepherd, Jan 30 2013

A228721 Decimal expansion of 7*Pi.

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Oct 03 2013

7*Pi is also the surface area of a sphere whose diameter equals the square root of 7. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013

			21.99114857512855266923850368295652018938...

Index entries for transcendental numbers

Cf. A000796, A019692, A122952, A019694, A019669, A228719 (Pi through 6*Pi).

RealDigits[7*Pi,10,120][[1]] (* Harvey P. Dale, Jan 14 2015 *)

7*Pi \\ Charles R Greathouse IV, Oct 01 2022

A232273 Decimal expansion of the arctan of Pi.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Nov 22 2013

			1.2626272556789116834443220836056983435089476704243835969738099522522...

Cf. A000796, A019669, A232182, A232272.

Mathematica
```
RealDigits[ArcTan[Pi], 10, 90][[1]]
```

atan(Pi) \\ Charles R Greathouse IV, Nov 26 2024

Equals A019669 - A232272.

cp's OEIS Frontend

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

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A197762 Decimal expansion of sqrt(1/phi), where phi = (1 + sqrt(5))/2 is the golden ratio.

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A232247 Decimal expansion of the arctan of 2/Pi.

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A257817 Decimal expansion of the real part of li(i), i being the imaginary unit.

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A257818 Decimal expansion of the imaginary part of li(i), i being the imaginary unit.

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A352125 Decimal expansion of Pi*sqrt(2)*sqrt(2 + sqrt(2))/8.

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A378354 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a (small) triakis octahedron.

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A352125 Decimal expansion of Pisqrt(2)sqrt(2 + sqrt(2))/8.