A232736 -id:A232736 - cp's OEIS frontend

A255241 Decimal expansion of 2cos(3Pi/7).

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

Offset: 0

Wolfdieter Lang, Mar 13 2015

This is also the decimal expansion of 2*sin(Pi/14).
rho_2 := 2*cos(3*Pi/7) and rho(7) := 2*cos(Pi/7) (see A160389) are the two positive zeros of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of the algebraic number rho(7), the length ratio of the smaller diagonal and the side in the regular 7-gon (heptagon). See A187360 and a link to the arXiv paper given there, eq. (20) for the zeros of C(n, x). The other zero is negative, rho_3 := 2*cos(5*Pi/n). See -A255249.
Also the edge length of a regular 14-gon with unit circumradius. Such an m-gon is not constructible using a compass and a straightedge (see A004169). With an even m, in fact, it would be constructible only if the (m/2)-gon were constructible, which is not true in this case (see A272487). - Stanislav Sykora, May 01 2016

			0.445041867912628808577805128993589518932711137529089910623974031...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 3

Cf. A004169, A160389, A187360, A255249, A232736, A255240.

Edge lengths of other nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A130880 (n=18), A272491 (n=19). - Stanislav Sykora, May 01 2016

R:= RealField(120); 2*Cos(3*Pi(R)/7); // G. C. Greubel, Sep 04 2022

RealDigits[N[2Cos[3Pi/7], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/14)
```

polrootsreal(x^3 - x^2 - 2*x + 1)[2] \\ Charles R Greathouse IV, Oct 30 2023

numerical_approx(2*cos(3*pi/7), digits=120) # G. C. Greubel, Sep 04 2022

2*cos(3*Pi/7) = 2*sin(Pi/14) = 2*A232736 = 1/A231187 = 0.4450418679...
See A232736 for the decimal expansion of cos(3*Pi/7) = sin(Pi/14).
Equals i^(6/7) - i^(8/7). - Peter Luschny, Apr 04 2020
From Peter Bala, Oct 11 2021: (Start)
Equals 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)), where z = exp(2*Pi*i/7).
Equals 1 - A255240. (End)

Offset corrected by Stanislav Sykora, May 01 2016

A073052 Decimal expansion of cos(Pi/7).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 15 2002

Cubic number with denominator 2 and minimal polynomial 8*x^3 - 4*x^2 - 4*x + 1. - Charles R Greathouse IV, May 13 2019

			Cos(Pi/7) = 0.90096886790241912623610231950744505116139...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Kevin S. Brown's MathPages, Radical Expression For cos(2*Pi/7)
Index entries for algebraic numbers, degree 3.

Cf. A232736 (cos(3*Pi/7)).

R:= RealField(120); Cos(Pi(R)/7); // G. C. Greubel, Sep 04 2022

Mathematica
```
RealDigits[ Cos[Pi/7], 10, 105] [[1]]
```

cos(Pi/7) \\ Charles R Greathouse IV, May 13 2019

numerical_approx(cos(pi/7), digits=120) # G. C. Greubel, Sep 04 2022

Equals Hypergeometric2F1([5/7, 2/7], [1/2], 3/4)/2 . - R. J. Mathar, Oct 27 2008

Equals (1/6)*(1 + (7*(-1 +3*sqrt(3)*I)/2)^(1/3) + (-7*(1 +3*sqrt(3)*I)/2)^(1/3)). - G. C. Greubel, Sep 04 2022

A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232736.

Root of the equation -7 + 56*x^2 - 112*x^4 + 64*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.974927912181823607018131682993931217232785800619997437648...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
A. Arman, A. Bondarenko, and A. Prymak, Convex bodies of constant width with exponential illumination number, arXiv:2304.10418 [math.MG], 2023.
Index entries for algebraic numbers, degree 6

Cf. A232736 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232737 (real(I^(1/8))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

R:= RealField(100); Cos(Pi(R)/14); // G. C. Greubel, Sep 19 2022

RealDigits[Cos[Pi/14],10,120][[1]] (* Harvey P. Dale, Dec 15 2018 *)

numerical_approx(cos(pi/14), digits=120) # G. C. Greubel, Sep 19 2022

2*this^2 -1 = A073052. - R. J. Mathar, Aug 29 2025

Equals 2F1(-1/14,1/14;1/2;1) . - R. J. Mathar, Aug 31 2025

A280533 Decimal expansion of 14*sin(Pi/14).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 04 2017

Decimal expansion of the ratio of the perimeter of a regular 7-gon (heptagon) to its diameter (largest diagonal).

			3.115293075388401660044635902955126632528977962703629374367818222563897248...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280585 (n=8), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A232736.

RealDigits[14*Sin[Pi/14], 10, 129][[1]] (* G. C. Greubel, Sep 20 2022 *)

PARI
```
14*sin(Pi/14)
```

numerical_approx(14*sin(pi/14), digits=127) # G. C. Greubel, Sep 20 2022

Equals 14*A232736.

A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232737.

			0.195090322016128267848284868477022240927691617751954807754502...

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

Cf. A232737 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232736 (imag(I^(1/7))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

R:= RealField(116); Sin(Pi(R)/16); // G. C. Greubel, Sep 20 2022

RealDigits[Sin[Pi/16],10,120][[1]] (* Harvey P. Dale, Sep 01 2018 *)

imag(I^(1/8)) \\ Seiichi Manyama, Apr 04 2021

sin(Pi/16) \\ Seiichi Manyama, Apr 04 2021

sqrt(2-sqrt(2+sqrt(2)))/2 \\ Seiichi Manyama, Apr 04 2021

numerical_approx(sin(pi/16), digits=116) # G. C. Greubel, Sep 20 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
This^2 + A232737^2 = 1.
Smallest positive of the 8 real-valued roots of 128*x^8-256*x^6+160*x^4-32*x^2+1=0.

A323601 Decimal expansion of sin(Pi/7).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 19 2019

			0.43388373911755812047576833284835875460999072778745987644454730353220325...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A019829 (sin(Pi/9)), A232736 (sin(Pi/14)).

Cf. A121598, A272487.

SetDefaultRealField(RealField(100)); R:= RealField(); Sin(Pi(R)/7); // G. C. Greubel, Feb 08 2019

Mathematica
```
RealDigits[Sin[Pi/7], 10, 120][[1]]
```

default(realprecision, 100); sin(Pi/7) \\ G. C. Greubel, Feb 08 2019

polrootsreal(64*x^6-112*x^4+56*x^2-7)[4] \\ Charles R Greathouse IV, Feb 05 2025

numerical_approx(sin(pi/7), digits=100) # G. C. Greubel, Feb 08 2019

Root of the equation 64*x^6 - 112*x^4 + 56*x^2 - 7 = 0. (Other +- A232735 and +- 0.7818314... = +- cos(3*Pi/14))
Equals sqrt((196 + 7*i*2^(2/3)*(21*i*sqrt(3) - 7)^(1/3)*(i + sqrt(3)) + i*2^(4/3)*(21*i*sqrt(3) - 7)^(2/3)*(2*i + sqrt(3)))/336), where i is the imaginary unit.
Equals cos(5*Pi/14).
From Gleb Koloskov, Jul 15 2021: (Start)
Positive root of the equation x^3 + sqrt(7)/2*x^2 - sqrt(7)/8 = 0.
Equals ((4*sqrt(7)*(13+3*sqrt(3)*i))^(1/3)+28*(4*sqrt(7)*(13+3*sqrt(3)*i))^(-1/3)-2*sqrt(7))/12, where i is the imaginary unit. (End)
Equals 1/A121598 = A272487/2. - Hugo Pfoertner, Dec 15 2024
This^2 + A073052^2=1. - R. J. Mathar, Aug 31 2025

A343059 Decimal expansion of tan(Pi/14).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

Root of the equation -1 + 21*x^2 - 35*x^4 + 7*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.228243474390149938077611362061014782...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A232736 (sin(Pi/14)), A232735 (cos(Pi/14)).

RealDigits[Tan[Pi/14], 10, 125][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/14)
```

numerical_approx(tan(pi/14), digits=124) # G. C. Greubel, Sep 30 2022

A362922 Decimal expansion of cos(2Pi/7) = sin(3Pi/14) = A255249/2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 25 2023

This number, negated, is a zero of the polynomial 8*x^3 - 4*x^2 - 4*x + 1 that arises in the dissection of a regular heptagon. The other two zeros are cos(Pi/7) (A073052) and sin(Pi/14) (A232736).

The old definition was: Decimal expansion of 1/(8*cos(Pi/7)*sin(Pi/14)).

			0.6234898018587335305250048840042398106322747308964021053655...

Cf. A073052, A160389, A232736, A255249.

Digits := 110: evalf(((-1)^(2/7) - (-1)^(5/7))/2, Digits)*10^96:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Jun 25 2023

First@ RealDigits[1/(8*Cos[Pi/7]*Sin[Pi/14]), 10, 96] (* Michael De Vlieger, Jun 25 2023 *)

Equals 1/(4*cos(Pi/7)-2) = 1/(2*A160389-2). - Alois P. Heinz, Jun 25 2023

Equals (i^(4/7) - i^(10/7))/2. - Peter Luschny, Jun 26 2023

Simpler definition from Alois P. Heinz, Jun 25 2023.

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

			0.09801714032956060199419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
imag(I^(1/16))
```
PARI
```
sin(Pi/32)
```
PARI
```
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).

One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025

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A255241 Decimal expansion of 2*cos(3*Pi/7).

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A073052 Decimal expansion of cos(Pi/7).

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A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

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A280533 Decimal expansion of 14*sin(Pi/14).

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A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

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A323601 Decimal expansion of sin(Pi/7).

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A255241 Decimal expansion of 2cos(3Pi/7).

A362922 Decimal expansion of cos(2Pi/7) = sin(3Pi/14) = A255249/2.