A375067 -id:A375067 - cp's OEIS frontend

A019952 Decimal expansion of tangent of 54 degrees.

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

A375068 Decimal expansion of the sagitta of a regular pentagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 29 2024

			0.1624598481164531630779357061075672324774517357607...

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

Cf. A300074 (circumradius), A375067 (apothem), A102771 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375069 (hexagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A019916, A179050, A229760.

First[RealDigits[Tan[Pi/10]/2, 10, 100]]

tan(Pi/10)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(80*x^4-40*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/10)/2 = sqrt(1-2/sqrt(5))/2 = A019916/2.
Equals A300074 - A375067.
Equals A179050/5 = sqrt(A229760)/10. - Hugo Pfoertner, Jul 30 2024

A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 26 2024

			1.0382606982861682835817694307429201653528601033...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 6.

Cf. A374957 (circumradius), A374972 (sagitta), A178817 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A178818, A073052, A343058.

Mathematica
```
First[RealDigits[Cot[Pi/7]/2, 10, 100]]
```

 5/(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/7)/2 = A178818/2.
Equals 1/(2*tan(Pi/7)) = 1/(2*A343058).
Equals A374957*cos(Pi/7) = A374957*A073052.
Equals A374957 - A374972.
Largest of the 6 real-valued roots of 448*x^6 -560*x^4 +84*x^2 -1 =0. - R. J. Mathar, Aug 29 2025

A375152 Decimal expansion of the apothem (inradius) of a regular 9-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2024

			1.3737387097273111393808320132488363588759362995854...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 6.

Cf. A375151 (circumradius), A375153 (sagitta), A256853 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A019879, A019918, A019968.

Mathematica
```
First[RealDigits[Cot[Pi/9]/2, 10, 100]]
```

 5/tan(Pi/9) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/9)/2 = A019968/2.
Equals 1/(2*tan(Pi/9)) = 1/(2*A019918).
Equals A375151*cos(Pi/9) = A375151*A019879.
Equals A375151 - A375153.
Largest of the 6 real-valued roots of 192*x^6 -432*x^4 +132*x^2 -1=0. - R. J. Mathar, Aug 29 2025

A375191 Decimal expansion of the apothem (inradius) of a regular 11-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 04 2024

			1.702843619444625004524065173324424415978649993...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 10.

Cf. A375190 (circumradius), A375192 (sagitta), A256854 (area).

Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375193 (12-gon).

First[RealDigits[Cot[Pi/11]/2, 10, 100]]

 5/tan(Pi/11) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/11)/2.
Equals 1/(2*tan(Pi/11)).
Equals A375190*cos(Pi/11).
Equals A375190 - A375192.

A375193 Decimal expansion of the apothem (inradius) of a regular 12-gon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 04 2024

Apart from the first digit the same as A010527.

			1.8660254037844386467637231707529361834714026269...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 2.

Cf. A188887 (circumradius), A375194 (sagitta), A178809 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon).
Cf. A019884, A019913, A019973, A332133, A375069.

First[RealDigits[Cot[Pi/12]/2, 10, 100]]

sqrt(3)/2 + 1 \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/12)/2 = (2 + sqrt(3))/2 = A019973/2.
Equals 1/(2*tan(Pi/12)) = 1/(2*A019913).
Equals A188887*cos(Pi/12) = A188887*A019884.
Equals A188887 - A375194.
Equals A332133^2 = 2 - A375069. - Hugo Pfoertner, Aug 04 2024

A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 06 2024

			72.60725303413392187893153397383948620117264765443...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 8.

Cf. A377751 (volume), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A010527 (analogous for a regular icosahedron, with offset 1).
Cf. A002163, A002194, A375067.

First[RealDigits[3*(10*Sqrt[3] + Sqrt[25 + Sqrt[500]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "SurfaceArea"], 10, 100]]

3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) \\ Charles R Greathouse IV, Feb 05 2025

Equals 3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) = 30*A002194 + 3*sqrt(25 + 10*A002163).

Equals 30*(A002194 + A375067).

cp's OEIS Frontend

A019952 Decimal expansion of tangent of 54 degrees.

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A375068 Decimal expansion of the sagitta of a regular pentagon with unit side length.

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A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

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A375152 Decimal expansion of the apothem (inradius) of a regular 9-gon with unit side length.

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A375191 Decimal expansion of the apothem (inradius) of a regular 11-gon with unit side length.

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A375193 Decimal expansion of the apothem (inradius) of a regular 12-gon with unit side length.

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A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

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