A019845 -id:A019845 - cp's OEIS frontend

A019881 Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).

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

Offset: 0

N. J. A. Sloane

Circumradius of pentagonal pyramid (Johnson solid 2) with edge 1. - Vladimir Joseph Stephan Orlovsky, Jul 19 2010
Circumscribed sphere radius for a regular icosahedron with unit edges. - Stanislav Sykora, Feb 10 2014
Side length of the particular golden rhombus with diagonals 1 and phi (A001622); area is phi/2 (A019863). Thus, also the ratio side/(shorter diagonal) for any golden rhombus. Interior angles of a golden rhombus are always A105199 and A137218. - Rick L. Shepherd, Apr 10 2017
An algebraic number of degree 4; minimal polynomial is 16x^4 - 20x^2 + 5, which has these smaller, other solutions (conjugates): -A019881 < -A019845 < A019845 (sine of 36 degrees). - Rick L. Shepherd, Apr 11 2017
This is length ratio of one half of any diagonal in the regular pentagon and the circumscribing radius. - Wolfdieter Lang, Jan 07 2018
Quartic number of denominator 2 and minimal polynomial 16x^4 - 20x^2 + 5. - Charles R Greathouse IV, May 13 2019
This gives the imaginary part of one of the members of a conjugate pair of roots of x^5 - 1, with real part (-1 + phi)/2 = A019827, where phi = A001622. A member of the other conjugte pair of roots is (-phi + sqrt(3 - phi)*i)/2 = (-A001622 + A182007*i)/2 = -A001622/2  + A019845*i. - Wolfdieter Lang, Aug 30 2022

			0.95105651629515357211643933337938214340569863412575022244730564443015317008...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Golden Rhombus.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Wolfram Alpha, Johnson solid 2.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827, A102208, A179290 (inverse), A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A182007.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A179296 (dodecahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

SetDefaultRealField(RealField(100)); Sqrt((5 + Sqrt(5))/8); // G. C. Greubel, Nov 02 2018

Digits:=100: evalf(sin(2*Pi/5)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sqrt[(5 + Sqrt[5])/8], 10, 111]  (* Robert G. Wilson v *)
RealDigits[Sin[2 Pi/5], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

default(realprecision, 120);
real(I^(1/5)) \\ Rick L. Shepherd, Apr 10 2017

Equals sqrt((5+sqrt(5))/8) = cos(Pi/10). - Zak Seidov, Nov 18 2006
Equals 2F1(13/20,7/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals the real part of i^(1/5). - Stanislav Sykora, Apr 25 2012
Equals A001622*A182007/2. - Stanislav Sykora, Feb 10 2014
Equals sin(2*Pi/5) = sqrt(2 + phi)/2 = -sin(3*Pi/5), with phi = A001622  - Wolfdieter Lang, Jan 07 2018
Equals 2*A019845*A019863. - R. J. Mathar, Jan 17 2021

A182007 Decimal expansion of 2*sin(Pi/5).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 06 2012

The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).
Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.
Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014
This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016
rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.

			1.1755705045849462583374119...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pentagon.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A002193, A003401, A019881, A019845, A063226, A102769, A131595, A188593, A300074.

SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

evalf(2*sin(Pi/5),100); # Muniru A Asiru, Nov 02 2018

RealDigits[2*Sin[Pi/5],10,120][[1]] (* Harvey P. Dale, Sep 29 2012 *)

2*sin(Pi/5) \\ Stanislav Sykora, May 02 2016

Equals sqrt(3-phi).
Equals sqrt((5-sqrt(5))/2). - Jean-François Alcover, May 21 2013
Equals Product_{k>=0} ((10*k + 4)*(10*k + 6))/((10*k + 3)*(10*k + 7)). - Antonio Graciá Llorente, Mar 25 2024
Equals Product_{k>=1} (1 - (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A019845 = 1/A300074. - Hugo Pfoertner, Nov 23 2024

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Decimal expansion of cos(2*Pi/5) (angle of 72 degrees).
Also the imaginary part of i^(1/5). - Stanislav Sykora, Apr 25 2012
One of the two roots of 4x^2 + 2x - 1 (the other is the sine of 54 degrees times -1). - Alonso del Arte, Apr 25 2015
This is the height h of the isosceles triangle in a regular pentagon inscribed in a unit circle, formed by a diagonal as base and two adjacent radii. h = cos(2*Pi/5) = sin(Pi/10). - Wolfdieter Lang, Jan 08 2018
Quadratic number of denominator 2 and minimal polynomial 4x^2 + 2x - 1. - Charles R Greathouse IV, May 13 2019
Largest superstable width of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			0.30901699437494742410229341718281905886015458990288143106772431135263...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.9 and 8.19, pp. 66, 535.

Zak Seidov, Table of n, a(n) for n = 0..999
Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019845, A019863, A055588, A094214, A134945, A187798, A341332, A377697.

RealDigits[Sin[18 Degree], 10, 108][[1]] (* Alonso del Arte, Apr 20 2015 *)

sin(Pi/10) \\ Charles R Greathouse IV, Feb 03 2015

polrootsreal(4*x^2 + 2*x - 1)[2] \\ Charles R Greathouse IV, Feb 03 2015

Equals (sqrt(5) - 1)/4 = (phi - 1)/2 = 1/(2*phi), with phi from A001622.
Equals 1/(1 + sqrt(5)). - Omar E. Pol, Nov 15 2007
Equals 1/A134945. - R. J. Mathar, Jan 17 2021
Equals 2*A019818*A019890. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} 1 - 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
Equals Product_{k>=1} (1 - 1/A055588(k)). - Amiram Eldar, Nov 28 2024
Equals A094214/2 = 1-A187798 = A341332/Pi = (A377697-2)/3. - Hugo Pfoertner, Nov 28 2024
This^2 + A019881^2 = 1. - R. J. Mathar, Aug 31 2025

A019952 Decimal expansion of tangent of 54 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 36 degrees. - Mohammad K. Azarian, Jun 30 2013
A quartic number with denominator 5. - Charles R Greathouse IV, Aug 27 2017
Conjecture: Product (2/3) * (8/7) * (12/13) * (18/17) * (22/23) * (32/33) * ... * (a_n/b_n) = sqrt(25 + 10*sqrt(5))/5 = tan(3*Pi/10) = A019952, where a_n even, a_n + b_n = a(n), |a_n - b_n| = 1, n >= 0. - Dimitris Valianatos, Feb 14 2020
Also the limiting value of the distance between the lines F(n)*x + F(n+1)*y = 0 and F(n)*x + F(n+1)*y = F(n+2) (where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Apr 03 2021
Decimal expansion of the radius of an inscribed sphere in a rhombic triacontahedron with unit edge length. - Wesley Ivan Hurt, May 11 2021

			1.376381920471173538207209581910887679525899336...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Exact trigonometric constants.
Wikipedia, Rhombic triacontahedron.
Index entries for algebraic numbers, degree 4.

Cf. A019845, A019863, A019934, A090772, A375067.
Cf. A344171 (rhombic triacontahedron surface area).
Cf. A344172 (rhombic triacontahedron volume).
Cf. A344212 (rhombic triacontahedron midradius).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(3*Pi(R)/10); // G. C. Greubel, Nov 22 2018

Digits:=100: evalf(tan(3*Pi/10)); # Wesley Ivan Hurt, Oct 07 2014

RealDigits[Tan[3*Pi/10], 10, 100][[1]] (* Wesley Ivan Hurt, Oct 07 2014 *)
RealDigits[Tan[54 Degree],10,120][[1]] (* Harvey P. Dale, Jul 16 2016 *)

tan(3*Pi/10) \\ Charles R Greathouse IV, Aug 27 2017

from sympy import sqrt
[print(i, end=', ') for i in str(sqrt(1+2/sqrt(5)).n(110)) if i!='.'] # Karl V. Keller, Jr., Jun 19 2020

numerical_approx(tan(3*pi/10), digits=100) # G. C. Greubel, Nov 22 2018

Equals A019863/A019845 = 1/A019934. - R. J. Mathar, Jul 26 2010
The largest positive solution of cos(4*arctan(1/x)) = cos(6*arctan(1/x)). - Thomas Olson, Oct 03 2014
Equals sqrt(25 + 10*sqrt(5))/5. - G. C. Greubel, Nov 22 2018
Equals sqrt(2 + sqrt(5))/5^(1/4). - Burak Muslu, Apr 03 2021
From Wesley Ivan Hurt, May 11 2021: (Start)
Equals phi^2/sqrt(1+phi^2) where phi is the golden ratio.
Equals sqrt(1+2/sqrt(5)). (End)
Equals Product_{k>=1} (1 - (-1)^k/A090772(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A375067. - Hugo Pfoertner, Nov 23 2024

A121570 Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 08 2006

1 + csc(Pi/5) is the radius of the smallest circle into which 5 unit circles can be packed ("r=2.701+ Proved by Graham in 1968.", according to the Friedman link, which has a diagram).
csc(Pi/5) = 1/A019845 is the distance between the center of the larger circle and the center of each unit circle.
The problem of finding the diameter d of the circumscribing circle of a regular pentagon of side s = 10 (in some length units) appears as an example in Abū Kāmil's treatise on the pentagon and decagon (see the Havil reference) and Abū Kāmil links. The answer is d/s = 1/sin(Pi/5). - Wolfdieter Lang, Mar 01 2018
Longer diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018
The length of the longer side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022
The radius of a common circle surrounded by 5 tangent unit circles is A121570 - 1. - Thomas Otten, Dec 27 2023

			1.701301616704079864363080994126...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58.

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. Friedman, Erich's Packing Center: "Circles in Circles"
I. C. Karpinski, The Algebra of Abu Kamil, Amer. Math. Month. XXI,2 (1914), 37-48.
MacTutor History of Mathematics, Abu Kamil Shuja.
Eric Weisstein's World of Mathematics, Golden Rhombus
Wikipedia, Abu Kamil.
Index entries for algebraic numbers, degree 4

Cf. A001622, A019845 (inverse), A182007 (2/A121570).

Cf. A179290 (shorter golden rhombus diagonal).

SetDefaultRealField(RealField(100)); R:= RealField(); 1/Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

evalf(1/sin(Pi/5),130); # Muniru A Asiru, Nov 02 2018

RealDigits[Csc[Pi/5], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)

PARI
```
1/sin(Pi/5)
```

numerical_approx(1/sin(pi/5), digits=100) # G. C. Greubel, Dec 12 2018

Equals 1/A019845.
Equals 2*(2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = A001622. - Wolfdieter Lang, Mar 01 2018
Equals sqrt(2 + 2 / sqrt(5)). - Michal Paulovic, Sep 01 2022
The minimal polynomial is 5*x^4 - 20*x^2 + 16. - Joerg Arndt, Sep 09 2022

A019857 Decimal expansion of sine of 48 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.74314482...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

Cf. A019833, A019875.

RealDigits[ Sin[4 Pi/15], 10, 111] (* Robert G. Wilson v *)

sin(4*Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(7*pi/30) = 2F1(17/20,3/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2*A019833*A019875. - R. J. Mathar, Jan 17 2021
Equals 1/(sqrt(5+2*sqrt(5))-sqrt(3)). - Seiichi Manyama, Mar 19 2021
4*this^3 -3*this = -A019845. - R. J. Mathar, Aug 29 2025
One of the 8 real-valued roots of 256*x^8-448*x^6+224*x^4-32*x^2+1=0. - R. J. Mathar, Aug 31 2025

A019890 Decimal expansion of sine of 81 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the real part of i^(1/10). - Stanislav Sykora, Apr 25 2012
Equals sin(9*Pi/20). - Wesley Ivan Hurt, Sep 01 2014
An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.98768834059513772619004024769343726075840686158988043492390480163...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8

Digits:=100: evalf(sin(9*Pi/20)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sin[81 Degree],10,120][[1]] (* Harvey P. Dale, Aug 11 2012 *)

sin(9*Pi/20) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/20) = sqrt((1+A019881)/2) = sqrt(1-A019818^2) = sqrt(5-sqrt(5))*(sqrt(5)+sqrt(5+2*sqrt(5)))/(4*sqrt(5)). - R. J. Mathar, Jun 18 2006
Equals A019863 * A019872 + A019836 * A019845 = A019887 * A019896 + A019812 * A019821. - R. J. Mathar, Jan 27 2021
Root of 256*x^8 -512*x^6 +304*x^4 -48*x^2+1=0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/10,1/10;1/2;1/2). - R. J. Mathar, Aug 31 2025

A019934 Decimal expansion of tangent of 36 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 54 degrees. - Mohammad K. Azarian, Jun 30 2013

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			0.72654252800536088589546675748061874961609239296520...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 4.

Cf. A019934, A019952, A019970, A002163, A063226.

RealDigits[Tan[36 Degree],10,120][[1]] (* Harvey P. Dale, Nov 06 2012 *)

tan(Pi/5) \\ Charles R Greathouse IV, Aug 27 2017

This number is sqrt(5-2*sqrt(5)). This number * A019970 = sqrt(5) = A002163. - R. J. Mathar, Jun 18 2006
The smallest positive solution of cos(4*arctan(x)) = cos(6*arctan(x)). - Thomas Olson, Oct 03 2014
Let r(n) = (n - 1)/(n + 1) if n mod 4 = 1, (n + 1)/(n - 1) otherwise; then this constant (A019934) equals with Product_{n>=0} r(10*n+5) = (2/3) * (8/7) * (12/13) * (18/17) * ... - Dimitris Valianatos, Sep 14 2019
Equals Product_{k>=1} (1 + (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 1/A019952. - Hugo Pfoertner, Nov 23 2024
tan(Pi/5) = A019845 / A019863. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of x^4-10*x^2+5=0. (Other A019970). - R. J. Mathar, Aug 31 2025

A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Feb 25 2014

Equals phi^2/(2*xi), where phi is the golden ratio (A001622, 2*cos(Pi/5)) and xi is its associate (A182007, 2*sin(Pi/5)).

			1.1135163644116067351943750394869493758831503698864877726012080...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A182007, A019863, A019863, A019952, A374771 (sphere volume).

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A179294 (icosahedron).

RealDigits[ Cos[Pi/5]^2 / Sin[Pi/5], 10, 111][[1]] (* Or *)
RealDigits[ Sqrt[5/8 + 11/(8 Sqrt[5])], 10, 111][[1]] (* Robert G. Wilson v, Feb 28 2014 *)

PARI
```
sqrt(250+110*sqrt(5))/20
```

Equals A001622^2/A182007 = (cos(Pi/5))^2/sin(Pi/5) = A019863^2/A019845 = cos(Pi/5)*cotan(Pi/5) = A019863*A019952 = 1/sin(Pi/5) - sin(Pi/5) = A019845^(-1) - A019845 = sqrt(250+110*sqrt(5))/20.

A340722 Decimal expansion of Gamma(4/5).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.164229713725303373636...

DLMF, Gamma function properties, DLMF section 5.5.
Index to sequences related to gamma function.

Cf. A019845, A175380, A340721, A340723.

Maple
```
evalf(GAMMA(4/5),120) ;
```

RealDigits[Gamma[4/5], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A175380 = Pi/A019845. [DLMF (5.5.3)]

this * A340723 * 2^(1/10)/sqrt(2*Pi) = A340721. [DLMF (5.5.5)]

cp's OEIS Frontend

A019881 Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).

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A182007 Decimal expansion of 2*sin(Pi/5).

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A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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A019952 Decimal expansion of tangent of 54 degrees.

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A121570 Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).

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A019857 Decimal expansion of sine of 48 degrees.

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