A019684 -id:A019684 - 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

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

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.

A377968 Decimal expansion of tan(arctan(4)/4).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Nov 13 2024

			0.34415073140891080771475922788534684760565020832584...

Michael Penn, A beautiful identity, YouTube video, 2024.
Index entries for algebraic numbers, degree 4

Cf. A010473, A019684, A195628.

First[RealDigits[Tan[ArcTan[4]/4], 10, 100]]

Equals 2*(cos(6*Pi/17) + cos(10*Pi/17)) = 2*(cos(6*A019684) + cos(10*A019684)).
Equals (sqrt(34 + 2*sqrt(17)) - sqrt(17) - 1)/4 = (sqrt(34 + 2*A010473) - A010473 - 1)/4.
Equals the root closest to 0 of x^4 + x^3 - 6*x^2 - x + 1.

cp's OEIS Frontend

A210649 Decimal expansion of cos(Pi/17).

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A019700 Decimal expansion of 2*Pi/17.

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

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Mathematica

PARI

Formula

A377968 Decimal expansion of tan(arctan(4)/4).

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Formula