A210644 -id:A210644 - cp's OEIS frontend

A210649 Decimal expansion of cos(Pi/17).

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

Offset: 0

Bruno Berselli, Mar 27 2012

This algebraic number is related to the constructibility of the regular heptadecagon (see also A210644), it 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(Pi/17) is 0, 1, 57, 1, 2, 1, 2, 2, 8, 9, 2, 3, 1, 1, 1, 1, 1, 2, 2, 13, 5, 1, 7, 84, 1, 1, 1,...
Expressed in terms of radicals, cos(Pi/17) is (1/8)*sqrt(2*(2*sqrt(sqrt((17/2)*(17-sqrt(17))) - sqrt((1/2)*(17-sqrt(17))) - 4*sqrt(2*(17+sqrt(17))) + 3*sqrt(17) + 17) + sqrt(17) + sqrt(2*(17-sqrt(17))) + 15)). - Jean-François Alcover, Dec 21 2012

			cos(Pi/17) = 0.9829730996839017782819488448551987160987228750656328...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Heptadecagon.
Index entries for algebraic numbers, degree 8

Cf. A019684, A210644.

Mathematica
```
RealDigits[Cos[Pi/17], 10, 87][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(cos(%pi/17)));
```
PARI
```
cos(Pi/17)
```

Equals (i^(2/17) - i^(32/17))/2. - Peter Luschny, Apr 04 2020

A228787 Decimal expansion of 2*sin(Pi/17), the ratio side/R in the regular 17-gon inscribed in a circle of radius R.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Oct 07 2013

s(17) := 2*sin(Pi/17) is an algebraic integer of degree 16 (over the rationals). Its minimal polynomial is 17 - 204*x^2 + 714*x^4 - 1122*x^6 + 935*x^8 - 442*x^10 + 119*x^12 - 17*x^14 + x^16. Its coefficients in the power basis of the algebraic number field Q(2*cos(Pi/34)) are  [0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1] (see row l = 8 of A228785). The decimal expansion of 2*cos(Pi/34) is given in A228788.
The continued fraction expansion starts with  0; 2, 1, 2, 1, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 2, 1, 1, 43, 3, 1, 5, 2, 17, 2, ...
Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into s(17) = sqrt(2*(1 - cos(2*Pi/17))).
Since 17 is a Fermat prime, this number is constructible and can be written as an expression containing just integers, the basic four arithmetic operations, and square roots. See A003401 for more details. - Stanislav Sykora, May 02 2016

			0.367499035633140663148817679...

Cf. A003401, A019434, A210644, A228785, A228788.

RealDigits[2Sin[Pi/17], 10, 100][[1]] (* Alonso del Arte, Jan 01 2014 *)

2*sin(Pi/17) \\ Charles R Greathouse IV, Nov 12 2014

s(17) = 2*sin(Pi/17) = 2*A241243.

Equals sqrt(34-2*sqrt(17)-2*sqrt(34-2*sqrt(17))-4*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/4. - Stanislav Sykora, May 02 2016

Offset corrected by Rick L. Shepherd, Jan 01 2014

A019700 Decimal expansion of 2*Pi/17.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.3695991357164462633485462803858238687290787528676595....

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

Cf. A019684, A210644.

RealDigits[2 (Pi/17), 10, 99][[1]] (* Bruno Berselli, Mar 28 2012 *)

ev(2*%pi/17, numer); /* Bruno Berselli, Mar 28 2012 */

2*Pi/17  \\ Bruno Berselli, Mar 28 2012

A228788 Decimal expansion of the algebraic integer 2*cos(Pi/34) of degree 16 = A055034(34) (over the rationals), the length ratio (smallest diagonal)/side of a regular 34-gon.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Oct 07 2013

rho(34):= 2*cos(Pi/34) is used in the algebraic number field Q(rho(34)) of degree 16 (see A187360) in which s(17) = 2*cos(Pi/17) (for its decimal expansion see A228787), the length ratio side/R of a regular 17-gon inscribed in a circle of radius R, is an integer. See A228787 for this expansion.
Gauss' formula for cos(2*Pi/17), given in A210644, can be inserted into rho(34) = sqrt(2+sqrt(2+2*cos(2*Pi/17))).
The minimal polynomial of rho(34) is 17 - 204*x^2 + 714*x^4 - 1122*x^6 + 935*x^8 - 442*x^10 + 119*x^12 - 17*x^14 + x^16 (row n=34 polynomial of A187360).
The continued fraction expansion starts with 1; 1, 116, 4, 1, 2, 1, 20, 2, 2, 1, 7, 10, 2, 2, 1, 3, 6, 1, 4, 4, 15, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of the length ratio (smallest diagonal)/side of a regular 34-gon

Cf. A055034, A187360, A210644, A228787.

RealDigits[2 Cos[Pi/34], 10, 111][[1]] (* Robert G. Wilson v, Jul 28 2014 *)

2*cos(Pi/34) \\ Charles R Greathouse IV, Nov 12 2014

2*cos(Pi/34) = 1.99146835259006904374238235781096...

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.

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A210649 Decimal expansion of cos(Pi/17).

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A228787 Decimal expansion of 2*sin(Pi/17), the ratio side/R in the regular 17-gon inscribed in a circle of radius R.

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A019700 Decimal expansion of 2*Pi/17.

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A228788 Decimal expansion of the algebraic integer 2*cos(Pi/34) of degree 16 = A055034(34) (over the rationals), the length ratio (smallest diagonal)/side of a regular 34-gon.

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A303816 Decimal expansion of 2700/17.

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A303817 Decimal expansion of 360/17.

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

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