A019863 -id:A019863 - cp's OEIS frontend

A144981 Decimal expansion of cos(Pi/8) = cos(22.5 degrees).

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

Offset: 0

R. J. Mathar, Sep 28 2008

Also the real part of i^(1/4). - Stanislav Sykora, Apr 25 2012
Width of a regular octagon of unit diameter. See Bingane and Audet. - Michel Marcus, Oct 04 2021
Minimal polynomial 8x^4 - 8x^2 + 1. - Charles R Greathouse IV, Oct 30 2023
Also the ratio (1+sqrt(2))/sqrt(4+2*sqrt(2)) of the radii and perimeters of the inscribed and circumscribed circles of a regular octagon. This and the first two comments are actually all equivalent. - M. F. Hasler, Aug 13 2025

			Equals 0.923879532511286756128183189396788286822416625863642486115097...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Christian Bingane and Charles Audet, The equilateral small octagon of maximal width, arXiv:2110.00036 [math.MG], 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Octagon.
Index entries for algebraic numbers, degree 4

Cf. A019863: cos(Pi/5), A010527: cos(Pi/6), A073052: cos(Pi/7), A019879: cos(Pi/9).

Maple
```
evalf(sqrt(2+sqrt(2))/2) ;
```

RealDigits[ Sqrt[2 + Sqrt[2]]/2, 10, 111][[1]] (* Or *) RealDigits[ Cos[Pi/8], 10, 111][[1]] (* Robert G. Wilson v *)

cos(Pi/8) \\ Michel Marcus, Dec 15 2015

from math import isqrt # integer arithmetic, avoiding 10^(4N) in inner isqrt
def A144981_first(N=99): return [9] if N<2 else list(map(int,str(
    isqrt(isqrt(100**(N+2)>>3)*10**(N-2)+100**N//2)))) # M. F. Hasler, Aug 13 2025

numerical_approx(sqrt(2+sqrt(2))/2, digits=120) # G. C. Greubel, Sep 04 2022

Equals sqrt(2 + sqrt(2))/2 = sqrt(3.41421...)/2 = 1.8477759.../2.
Equals Hypergeometric2F1([11/16, 5/16], [1/2], 3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2F1(-1/4,1/4;1/2;1/2) . - R. J. Mathar, Aug 31 2025

A239798 Decimal expansion of the midsphere radius in a regular dodecahedron with unit edges.

Original entry on oeis.org

1, 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

Offset: 1

Stanislav Sykora, Mar 27 2014

In a regular polyhedron, the midsphere is tangent to all edges.

Apart from leading digits the same as A019863 and A019827. - R. J. Mathar, Mar 30 2014

			1.30901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A000045, A001622, A001654, A019827, A019863, A066983, A094884, A104457, A187799.

Midsphere radii in Platonic solids: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A019863 (icosahedron).

Digits:=100: evalf((3+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[GoldenRatio^2/2,10,105][[1]] (* Vaclav Kotesovec, Mar 27 2014 *)

PARI
```
(3+sqrt(5))/4
```

Equals phi^2/2, phi being the golden ratio (A001622).
Equals (3+sqrt(5))/4.
Equals lim_{n->oo} A000045(n)/A066983(n). - Dimitri Papadopoulos, Nov 23 2023
Equals Product_{k>=2} (1 + (-1)^k/A001654(k)). - Amiram Eldar, Dec 02 2024
Equals A094884^2 = A104457/2 = 10/A187799. - Hugo Pfoertner, Dec 02 2024

A019887 Decimal expansion of sine of 78 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Equals sin(13*Pi/30). - Wesley Ivan Hurt, Aug 31 2014

A quartic number with denominator 2 and minimal polynomial 16x^4 + 8x^3 - 16x^2 - 8x + 1. - Charles R Greathouse IV, Aug 27 2017

			0.9781476007338056379285667478695995324597378088626771078851...

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

Digits:=100: evalf(sin(13*Pi/30)); # Wesley Ivan Hurt, Aug 31 2014

First@RealDigits[Sin[78 Degree], 10, 120] (* Michael De Vlieger, Aug 31 2014 *)

cos(Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/15) = [sqrt(5)-1]*[1+sqrt(3)*sqrt{5+2*sqrt(5)}]/8 = [A002163-1]*[1+A002194*A019970]/8. - R. J. Mathar, Jun 18 2006
Equals 2*A019848*A019860. - R. J. Mathar, Jan 17 2021
4*this^3 -3*this = A019863. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/10,1/10 ; 1/2 ; 3/4). - R. J. Mathar, Aug 31 2025
A root of 16*x^4+8*x^3-16*x^2-8*x+1=0. - R. J. Mathar, Aug 31 2025

A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

Convergents are 4/2, 8/8, 32/24, 96/80, 320/256, 1024/832, 3328/2688, 10752/8704, 34816/28160, 112640/91136, 364544/294912, 1179648/954368, 3817472/3088384, 12353536/9994240, ... = A209084/A063727. - Seiichi Kirikami, Mar 14 2012

2*(-1 + phi) is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Feb 16 2016

			1.236067977499789696...

Michael Penn, How large is the blue ▲, YouTube video, 2021.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019863, A063727, A209084, A033887.

RealDigits[ N[4/(1+Sqrt[5]), 150] ] [ [1] ] (* Seiichi Kirikami, Mar 14 2012 *)

4/(1+sqrt(5)) \\ Altug Alkan, Apr 11 2016

Equals A134945 - 2 = A002163 - 1 = A098317 - 3. - R. J. Mathar, Oct 27 2008
2*(-1 + A001622). - Wolfdieter Lang, Feb 17 2016
Equals the harmonic mean of 1 and phi, 2*phi/(1+phi). - Stanislav Sykora, Apr 11 2016
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!-8*n!^2)/(n!^2*3^(2*n+2)).
Equals -1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)
Equals 1/A019863. - R. J. Mathar, Jan 17 2021
Equals 2*sin(Pi/5)/sin(2*Pi/5) = hypergeom([1/5, 3/5], [7/5], 1) = hypergeom([-1/5, -3/5], [3/5], 1). - Peter Bala, Mar 04 2022

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.

A375067 Decimal expansion of the apothem (inradius) of a regular pentagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 29 2024

			0.688190960235586769103604790955443839762949668...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 4.

Cf. A300074 (circumradius), A375068 (sagitta), A102771 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A019934, A019952, A019863, A131595.

Mathematica
```
First[RealDigits[Cot[Pi/5]/2, 10, 100]]
```

 5/tan(Pi/5) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/5)/2 = A019952/2.
Equals 1/(2*tan(Pi/5)) = 1/(2*A019934).
Equals sqrt(1/4 + 1/(2*sqrt(5))).
Equals (1/2)*csc(Pi/5)*cos(Pi/5) = A300074*A019863.
Equals A300074 - A375068.
Equals A131595/30. - Hugo Pfoertner, Jul 30 2024

A134944 Decimal expansion of (1 + sqrt(5))/8, the golden ratio divided by 4.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Nov 15 2007

Area of quadrilateral with side lengths (sqrt(5)+1)/4, (sqrt(5)-1)/4, 1/2, 1. [And what angles? - N. J. A. Sloane, Apr 28 2024]

			0.404508497187...

Index entries for algebraic numbers, degree 2.

Cf. A001622, A019863.

Digits:=100: evalf((1+sqrt(5))/8); # Wesley Ivan Hurt, Sep 08 2014

RealDigits[(1 + Sqrt[5])/8, 10, 100][[1]] (* Wesley Ivan Hurt, Sep 08 2014 *)

a(n)=floor(10^n*(1+sqrt(5))/8)%10 \\ Edward Jiang, Sep 07 2014

Edited by N. J. A. Sloane, Apr 28 2024

A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

Original entry on oeis.org

1, 1, 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, 7, 4, 2

Offset: 2

Jean-François Alcover, Jul 01 2014

Essentially the same digit sequence as A239798, A019863 and A019827. - R. J. Mathar, Jul 03 2014

The minimal polynomial of this constant is x^2 - 11*x - 1. - Joerg Arndt, Jan 01 2017

			11.09016994374947424102293417182819058860154589902881431067724311352630...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 83.

Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
Wikipedia, Hard Hexagon Model.
D. W. Wood and R. W. Turnbull, z^2-11z-1 as an algebraic invariant for the hard-hexagon model, 1988 J. Phys. A: Math. Gen. 21 L989.

Cf. A019863, A019827, A085850, A239798.

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458 A176522, A176537 (metallic means).

RealDigits[GoldenRatio^5, 10, 103] // First

(5*sqrt(5)+11)/2 \\ Charles R Greathouse IV, Aug 10 2016

Equals ((1 + sqrt(5))/2)^5 = (11 + 5*sqrt(5))/2.
Equals phi^5 = 11 + 1/phi^5 = 3 + 5*phi, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Nov 11 2023
Equals lim_{n->infinity} S(n, 5*(-1 + 2*phi))/ S(n-1, 5*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

cp's OEIS Frontend

A144981 Decimal expansion of cos(Pi/8) = cos(22.5 degrees).

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A239798 Decimal expansion of the midsphere radius in a regular dodecahedron with unit edges.

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A019887 Decimal expansion of sine of 78 degrees.

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A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

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A019890 Decimal expansion of sine of 81 degrees.

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A019934 Decimal expansion of tangent of 36 degrees.

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

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