A241243 -id:A241243 - cp's OEIS frontend

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

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

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
Equals A232738/(2*A343056). - R. J. Mathar, Sep 05 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

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

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A343061 Decimal expansion of tan(Pi/17).

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

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