A210649 -id:A210649 - cp's OEIS frontend

A210644 Decimal expansion of cos(2*Pi/17).

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

Offset: 0

Bruno Berselli, Mar 26 2012

Constant related to the constructibility of the regular heptadecagon. The "Disquisitiones Arithmeticae" of Gauss contains the following equivalent expression:
-1/16+(1/16)*sqrt(17)+(1/16)*sqrt(34-2*sqrt(17))+(1/8)*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt((34+2*sqrt(17)))).
This value is a root of the polynomial 256*x^8+128*x^7-448*x^6-192*x^5+240*x^4+80*x^3-40*x^2-8*x+1.
The continued fraction expansion of cos(2*Pi/17) is 0, 1, 13, 1, 4, 4, 2, 1, 1, 2, 4, 425, 1, 2, 5, 3, 1, 1, 1, 1, 1, 4, 4, 10, 3, 2, 1,...

			cos(2*Pi/17) = 0.9324722294043558045731158918215633862625877779451169...

C. F. Gauss, Disquisitiones Arithmeticae, 1801 (Lipsia), p. 662 (par. 365).
Ian Stewart, Professor Stewart's Cabinet of Mathematical Curiosities, BASIC Books, a member of the Perseus Books Group, NY, 2009, "Why Gauss Became a Mathematician", pp. 146 - 149.
Ian Stewart, Why Beauty Is Truth, A History of Symmetry, BASIC Books, a member of the Perseus Books Group, NY 2007, pp. 136.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Brady Haran and David Eisenbud, Heptadecagon and Fermat Primes (the math bit), Numberphile YouTube video, 2015.
Eric Weisstein's World of Mathematics, Heptadecagon.
Wikipedia, Heptadecagon.
Index entries for algebraic numbers, degree 8.

Cf. A019700, A210649.

RealDigits[Cos[2Pi/17], 10, 105][[1]]
RealDigits[(-1 + Sqrt[17] + Sqrt[34 - 2 Sqrt[17]] + Sqrt[68 + 12 Sqrt[17] - 4 Sqrt[170 + 38 Sqrt[17]]])/16, 10, 111][[1]] (* Robert G. Wilson v, Aug 09 2012 *)

Maxima
```
fpprec:90; ev(bfloat(cos(2*%pi/17)));
```
PARI
```
cos(2*Pi/17)
```

Equals (i^(4/17) - i^(30/17))/2. - Peter Luschny, Apr 04 2020

A019684 Decimal expansion of Pi/17.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.1847995678582231316742731401929119343645393764338297541....

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. A019700, A210649.

RealDigits[N[Pi/17,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

ev(%pi/17, numer); /* Bruno Berselli, Mar 28 2012 */

PARI
```
Pi/17 \\ Bruno Berselli, Mar 28 2012
```

A241243 Decimal representation of sin(Pi/17).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Apr 18 2014

			0.18374951781657033157440883962072758248913852384449940585065085774891492825305...

Saul Stahl, "Geometry From Euclid To Knots" Chapter 4.3, 'Regular Polygons', Courier - Dover Publications, NJ, 2009, pp. 153-154.

Wikipedia, Constructible polygon
Index entries for algebraic numbers, degree 16

Digits:=100: evalf(sin(Pi/17)); # Wesley Ivan Hurt, Aug 15 2014

RealDigits[ Sin[ Pi/17], 10, 111][[1]] (* Robert G. Wilson v, Aug 14 2014 *)
RealDigits[Root[17 - 816 x^2 + 11424 x^4 - 71808 x^6 + 239360 x^8 -
  452608 x^10 + 487424 x^12 - 278528 x^14 + 65536 x^16, 9],10,105][[1]] (* Artur Jasinski, Aug 04 2025 *)

Equals 1/4 sqrt(8 - sqrt(2*(15 + sqrt(17) - sqrt(2*(17 - sqrt(17))) + sqrt(2*(34 + 6 sqrt(17) + sqrt(2*(17 - sqrt(17))) - sqrt(34*(17 - sqrt(17))) + 8*sqrt(2*(17 + sqrt(17)))))))). - Robert G. Wilson v, Aug 14 2014
The minimal polynomial is 65536x^16 - 278528x^14 + 487424x^12 - 452608x^10 + 2398360x^8 - 71808 x^6 + 11424x^4 - 816x^2 + 17. - Robert G. Wilson v, Aug 14 2014
This^2 + A210649^2 = 1. - R. J. Mathar, Aug 31 2025

Offset corrected by Amiram Eldar, Aug 04 2025

A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.9951847266721968862448369...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 16

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022

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

PARI
```
real(I^(1/16))
```
PARI
```
cos(Pi/32)
```
PARI
```
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).
Satisfies 32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +84480*x^8 -21504*x^6 +2688*x^4 -128*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

A343061 Decimal expansion of tan(Pi/17).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

Root of the equation 17 - 680*x^2 + 6188*x^4 - 19448*x^6 + 24310*x^8 - 12376*x^10 + 2380*x^12 - 136*x^14 + x^16 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.18693239710797714594807628412307...

Index entries for algebraic numbers, degree 16

Cf. A241243 (sin(Pi/17)), A210649 (cos(Pi/17)).

RealDigits[Tan[Pi/17], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/17)
```

Equals sqrt((-2*(-8 + sqrt(2*(15 + sqrt(17) - sqrt(34 - 2*sqrt(17)) + sqrt(2*(34 + 6*sqrt(17) - sqrt(578 - 34*sqrt(17)) + sqrt(34 - 2*sqrt(17)) + 8*sqrt(2*(17 + sqrt(17)))))))))/(15 + sqrt(17) + sqrt(34 - 2*sqrt(17)) + sqrt(2*(34 + 6*sqrt(17) + sqrt(578 - 34*sqrt(17)) - sqrt(34 - 2*sqrt(17)) - 8*sqrt(2*(17 + sqrt(17))))))). - Vaclav Kotesovec, Apr 04 2021

A303816 Decimal expansion of 2700/17.

Original entry on oeis.org

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

Offset: 3

Omar E. Pol, Jun 13 2018

Decimal expansion of the internal angle of the regular heptadecagon (in degrees).

Period 16. - Jianing Song, Jun 22 2018

			158.82352941176470588235294117647058823529411764705882352941176470...

Wikipedia, Heptadecagon

Essentially the same as A007450. Cf. A019434, A210644, A210649, A303817.

Equals 180*15/17 = 158 + (14/17) = 180 - A303817.
From Chai Wah Wu, Dec 20 2019: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n > 12.
G.f.: x^3*(x^9 - 8*x^8 + 3*x^7 - 2*x^6 - x^5 + 6*x^4 - 3*x^2 - 4*x - 1)/((x - 1)*(x^8 + 1)). (End)

A303817 Decimal expansion of 360/17.

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Apr 30 2018

Decimal expansion of the external angle of the regular heptadecagon (in degrees).
The repeating pattern [1, 7, 6, 4, 7, 0, 5, 8, 8, 2, 3, 5, 2, 9, 4, 1] is the same as A007450. - Michael B. Porter, Jun 11 2018
Period 16. - Eric Chen, Jun 14 2018
Essentially the same as A021089. - R. J. Mathar, Aug 16 2018

			21.176470588235294117647058823529411764705882352941176470588235294...

Wikipedia, Heptadecagon

Cf. A007450, A019434, A210644, A210649, A303816.

RealDigits[360/17, 10, 111][[1]] (* Robert G. Wilson v, Jun 13 2018 *)

Equals 21 + (3/17) = 180 - A303816.
From Chai Wah Wu, Dec 20 2019: (Start)
a(n) = a(n-1) - a(n-8) + a(n-9) for n > 11.
G.f.: x^2*(-2*x^9 - 7*x^8 + 7*x^7 - 3*x^6 + 2*x^5 + x^4 - 6*x^3 + x - 2)/((x - 1)*(x^8 + 1)). (End)

A370393 Decimal expansion of the area of a unit heptadecagon (17-gon).

Original entry on oeis.org

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

Offset: 2

Michal Paulovic, Feb 17 2024

This constant multiplied by the square of the side length of a regular heptadecagon equals the area of that heptadecagon.
17^2 divided by this constant equals 68 * tan(Pi/17) = 12.71140300... which is the perimeter and the area of an equable heptadecagon with its side length 4 * tan(Pi/17) = 0.74772958... .
An equable rectangle with its perimeter and area = 17 has side lengths:
  a = s^2/8 = (17 - sqrt(17)) / 4 = (17 - A010473) / 4 = 3.21922359...
  b = 136/s^2 = (17 + sqrt(17)) / 4 = (17 + A010473) / 4 = 5.28077640...
  where s is the parameter from the formula mentioned below.

			22.7354918984165514...

Eric Weisstein's World of Mathematics, Heptadecagon.
Wikipedia, Heptadecagon.

Cf. A007450, A010473, A019684 (Pi/17), A210644 (cos(2*Pi/17)), A210649, A228787, A241243, A329592, A343061.

Maple
```
evalf(17 / (4 * tan(Pi/17)), 100);
```

RealDigits[17 / (4 * Tan[Pi/17]), 10, 100][[1]]

PARI
```
17 / (4 * tan(Pi/17))
```

Equals 17 / (4 * tan(Pi/17)) = 17 / (4 * A343061).
Equals 1 / (4 * A007450 * A343061).
Equals 17 * cos(Pi/17) / (4 * sin(Pi/17)).
Equals 17 * A210649 / (4 * A241243).
Equals 17 * A210649 / (2 * A228787).
Equals 17 * cot(Pi/17) / 4.
Equals 17 * sqrt(4 / (s^2 - 2 * s - 4 * sqrt(17 + 3 * sqrt(17) - s - sqrt(17) * 16/s)) - 1/16) where s = sqrt(34 - 2 * sqrt(17)) = 4 * A329592.
The minimal polynomial is 4294967296*x^16 - 3103113871360*x^14 + 510054948143104*x^12 - 28954726431195136*x^10 + 653743432704327680*x^8 - 6011468019822067712*x^6 + 20881180982314634240*x^4 - 21552361799603318912*x^2 + 2862423051509815793.

A387451 Decimal expansion of cos(Pi/34).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 29 2025

			0.99573417629503452187...

R. J. Mathar, Reduction formulas of the cosine of integer fractions of pi, (2008) vixra:2210.0026
Index entries for algebraic numbers, degree 16

Cf. A210649.

Equals sin(8*Pi/17)= sqrt( (A210649+1)/2 ).

Largest of the 16 real-valued roots of 65536*x^16 -278528*x^14 +487424*x^12 -452608*x^10 +239360*x^8 -71808*x^6 +11424*x^4 -816*x^2 +17 =0.

cp's OEIS Frontend

A210644 Decimal expansion of cos(2*Pi/17).

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A019684 Decimal expansion of Pi/17.

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A241243 Decimal representation of sin(Pi/17).

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A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

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A343061 Decimal expansion of tan(Pi/17).

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A303816 Decimal expansion of 2700/17.

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A303817 Decimal expansion of 360/17.

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