A121570 -id:A121570 - cp's OEIS frontend

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

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

A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Regular icosahedron: A three-dimensional figure with 20 congruent equilateral triangle faces, 12 vertices, and 30 edges.
Shorter diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018
The length of the shorter side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022
The side length of a square inscribed within a golden ellipse with a unit semi-major axis. - Amiram Eldar, Oct 02 2022
(10/3)*(this constant)=3.504874080794224... is the volume of the polyhedron with 32 edges with conjectured maximum volume inscribed in a sphere of radius 1. It has 60 congruent triangular faces and the symmetry group of the regular icosahedron. See Pfoertner links for visualizations. - Hugo Pfoertner, Aug 02 2025

			1.051462224238267212051338169695753214570995864486683563057871046482422...

Muniru A Asiru, Table of n, a(n) for n = 1..1000
J. Brandts, S. Korotov, M. Krizek, and J. Solc, On nonobtuse simplicial partitions, Siam Rev. 51 (2) (2009) 317-335.
Dr. Math, Regular Polyhedra: Formulas.
Hugo Pfoertner, Visualization of the 32-edge polyhedron with conjecturally maximum volume, (2021).
Hugo Pfoertner, Number of edges incident with the vertices of the 32-edge polyhedron, video (2021).
Eric Weisstein's World of Mathematics, Golden Rhombus.
Eric Weisstein's World of Mathematics, Icosahedron.
Wikipedia, Icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A010527, A019881, A090773, A102208.

Cf. A179290 (longer golden rhombus diagonal).

evalf[120](csc(2*Pi/5)); # Muniru A Asiru, Dec 11 2018

RealDigits[Csc[2 Pi/5], 10, 110][[1]] (* Eric W. Weisstein, Dec 11 2018 *)

sqrt(50-10*sqrt(5))/5 \\ Charles R Greathouse IV, Jan 22 2024

from decimal import *
getcontext().prec = 110
c = Decimal.sqrt(2 - 2 / Decimal.sqrt(Decimal(5)))
print([int(i) for i in str(c) if i != '.'])
# Karl V. Keller, Jr., Jul 10 2020

Equals sqrt(50-10*sqrt(5))/5.
Equals csc(2*Pi/5). - Eric W. Weisstein, Dec 11 2018
Equals 1/Im(e^(3*i*Pi/5)) = 1/Im(e^(3*i*Pi/5) - 1) = sqrt(2 - 2/sqrt(5)). - Karl V. Keller, Jr., Jun 11 2020
Equals 1/A019881. - R. J. Mathar, Jan 17 2021
From Antonio Graciá Llorente, Mar 15 2024: (Start)
Equals Product_{k >= 1} ((10*k - 1)*(10*k + 1))/((10*k - 2)*(10*k + 2)).
Equals Product_{k >= 1} 1/(1 - 1/(25*(2*k - 1)^2)). (End)
Equals Product_{k>=1} (1 - (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 2024
A root of 5*x^4 - 20*x^2 + 16=0 (see A121570). - R. J. Mathar, Aug 29 2025

Partially rewritten by Charles R Greathouse IV, Feb 02 2011

A121598 Decimal expansion of cosecant of 180/7 = 25.7142857+ degrees = csc(Pi/7).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 09 2006

1 + csc(Pi/7) is the radius of the smallest circle into which 8 unit circles can be packed ("r=3.304+ Proved by Braaksma in 1963.", according to the Friedman link, which has a diagram).

csc(Pi/7) is the distance between the center of the larger circle and the center of each unit circle that touches the larger circle.

			2.304764870962486505241150223546855...

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. Friedman, Erich's Packing Center: "Circles in Circles"
Index entries for algebraic numbers, degree 6.

Cf. A121570.

SetDefaultRealField(RealField(100)); R:= RealField(); 1/Sin(Pi(R)/7); // G. C. Greubel, Nov 02 2018

RealDigits[Csc[Pi/7], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)

PARI
```
1/sin(Pi/7)
```

Largest of the 6 real-values roots of 7*x^6 -56*x^4 +112*x^2 -64 =0. - R. J. Mathar, Aug 29 2025

A352324 Decimal expansion of 4Pi / (5sqrt(10-2*sqrt(5))).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 12 2022

Cauchy's residue theorem implies that Integral_{x=0..oo} 1/(1 + x^m) dx = (Pi/m) * csc(Pi/m); this is the case m = 5.

The area of a circle circumscribing a unit-area regular decagon.

			1.0689593321155951134251843725068826399014509252665...

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

Index entries for transcendental numbers.

Cf. A019694, A047209, A121570, A371604, A377405.

Integral_{x=0..oo} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), this sequence (m=5), A019670 (m=6), A352125 (m=8), A094888 (m=10).

evalf(4*Pi / (5*(sqrt(10-2sqrt(5)))), 100);

First[RealDigits[N[4Pi/(5Sqrt[10-2Sqrt[5]]), 93]]] (* Stefano Spezia, Mar 12 2022 *)

Equals Integral_{x=0..oo} 1/(1 + x^5) dx.
Equals (Pi/5) *csc(Pi/5).
Equals (1/2) * A019694 * A121570.
Equals 1/Product_{k>=1} (1 - 1/(5*k)^2). - Amiram Eldar, Mar 12 2022
Equals Product_{k>=2} (1 + (-1)^k/A047209(k)). - Amiram Eldar, Nov 22 2024
Equals 1/A371604 = A377405/5. - Hugo Pfoertner, Nov 22 2024

A121603 Numbers n such that the radius of the smallest circle into which n unit circles can be packed is 1 + csc(Pi/k), where k >= 2 is an integer.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11

Offset: 1

Rick L. Shepherd, Aug 09 2006

nonn

Corresponding k values are in A121604. For these n, the centers of k unit circles can form a regular k-gon with sides of length 2 centered at the center of the larger circle. From the diagrams in the link it appears likely that 13,18,19 are the next three terms.

			See A121602 for the case n=11 involving a 9-gon.

E. Friedman, Erich's Packing Center: "Circles in Circles"

Cf. A121604, A121570, A121598, A121601, A121602.

A121604 Numbers k such that the radius of the smallest circle into which A121603(m) unit circles can be packed is 1 + csc(Pi/k).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 09 2006

nonn

From the diagrams in the link it appears likely that 10,12,12 are the next three terms.

			See A121602 for the case a(8) = 9 pertaining to A121603(8) = 11 unit circles.

E. Friedman, Erich's Packing Center: "Circles in Circles"

Cf. A121603, A121570, A121598, A121601, A121602.

A356869 Decimal expansion of 4 / sqrt(5).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Sep 01 2022

The area of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			1.7888543819998317...

Cf. A121570, A179290, A204188.

cell2mat(struct2cell(struct(vpa(4 / sqrt(5), 105)))); ans(1:98)

parse(substring(convert(evalf(4 / sqrt(5), 105), string), 1..98));

RealDigits[4 / Sqrt[5], 10, 105][[1]][[Range[1, 97]]]

Equals [1; 1, 3, 1, 2] (periodic continued fraction expansion). - Peter Luschny, Sep 02 2022

Equals 1/A204188. - Alois P. Heinz, Sep 02 2022

A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Oct 10 2022

The perimeter of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			5.5055276818846941...

Cf. A019934, A019952, A019970, A121570, A179290, A204188, A356869.

Maple
```
sqrt(16 + 32 / sqrt(5));
```
Mathematica
```
Sqrt[16 + 32/Sqrt[5]]
```
PARI
```
sqrt(16 + 32 / sqrt(5))
```

Equals (4 / sqrt(5)) * sqrt(5 + 2 * sqrt(5)) = A356869 * A019970.
Equals sqrt(5 + 2 * sqrt(5)) / (sqrt(5) / 4) = A019970 / A204188.
Equals 4 * sqrt(1 + 2 / sqrt(5)) = 4 * A019952.
Equals 4 / sqrt(5 - 2 * sqrt(5)) = 4 / A019934.

A367480 Decimal expansion of the radius of a common circle surrounded by seven tangent unit circles.

Original entry on oeis.org

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

Offset: 1

Thomas Otten, Dec 23 2023

The radius of a common circle surrounded by n tangent unit circles (n > 2) is r = 1/sin(Pi/n) - 1.

n=7 is the smallest number for which the radius cannot be expressed using square roots, since the regular heptagon formed by the centers of the tangent circles is non-constructible (see A246724, A188582, and A121570 for n=3, 4, 5).

			1.3047648709624865052...

Andrew M. Gleason, Angle Trisection, the Heptagon, and the Triskaidecagon, The American Mathematical Monthly 95, no. 3 (March 1988), pp. 185-194.
Index entries for algebraic numbers, degree 6.

Cf. A121598.

Cf. A121570, A188582, A246724.

RealDigits[Csc[Pi/7] - 1, 10, 120][[1]] (* Amiram Eldar, Dec 28 2023 *)

PARI
```
1/sin(Pi/7) - 1
```

Equals 1 / sin(Pi/7) - 1.
Equals A121598 - 1.
Largest of the 6 real-valued roots of 7*x^6+ 42*x^5 +49*x^4 -84*x^3 -119*x^2 +42*x-1=0. - R. J. Mathar, Aug 29 2025

More digits from Jon E. Schoenfield, Dec 24 2023

Comments edited by Michal Paulovic, Dec 26 2023

A384851 Decimal expansion of minimal radius of a circle that contains 14 non-overlapping unit disks.

Original entry on oeis.org

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

Offset: 1

Jinyuan Wang, Jun 10 2025

			4.328428554860836681403909367478181...

Dinesh B. Ekanayake and Douglas J. LaFountain, Tight partitions for packing circles in a circle, Italian Journal of Pure and Applied Mathematics, 51 (2024), 115-136.
Michael Goldberg, Packing of 14, 16, 17 and 20 Circles in a Circle, Mathematics Magazine, Vol. 44, No. 3 (May, 1971), pp. 134-139 (provides D/d=4.3284 in Table 1, page 136).

Cf. A084618, A121570, A121598, A121601, A121602, A281115, A282279.

1+sqrt(1+polrootsreal(Pol([707281, -27270266, 472689975, -4930771548, 34449512067, -168736166642, 591369611109, -1498751280720, 2767422383674, -3746579404052, 3734397946902, -2743990597288, 1486108072662, -593729401364, 175537055738, -38557290064, 6295485573, -759438450, 66647843, -4134492, 172311, -4346, 49]))[10])

cp's OEIS Frontend

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

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A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.

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A121598 Decimal expansion of cosecant of 180/7 = 25.7142857+ degrees = csc(Pi/7).

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A352324 Decimal expansion of 4*Pi / (5*sqrt(10-2*sqrt(5))).

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A121603 Numbers n such that the radius of the smallest circle into which n unit circles can be packed is 1 + csc(Pi/k), where k >= 2 is an integer.

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A121604 Numbers k such that the radius of the smallest circle into which A121603(m) unit circles can be packed is 1 + csc(Pi/k).

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A356869 Decimal expansion of 4 / sqrt(5).

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A352324 Decimal expansion of 4Pi / (5sqrt(10-2*sqrt(5))).