A232738 -id:A232738 - cp's OEIS frontend

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232735.
Root of the equation 1 - 4*x - 4*x^2 + 8*x^3 = 0. - Vaclav Kotesovec, Apr 04 2021
The other 2 roots are -A362922 and A073052. - R. J. Mathar, Aug 29 2025

			0.222520933956314404288902564496794759466355568764544955311987...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 3.

Cf. A232735 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232738 (imag(I^(1/8))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232738.

			0.9807852804032304491261822361342390369739337308933360950029160885453...

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. A232738 (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))), A232735 (real(I^(1/7))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

RealDigits[Cos[Pi/16], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

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

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

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

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
Root of 128*x^8 -256*x^6 +160*x^4 -32*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
2*this^2 -1 = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8;1/2;1/2). - R. J. Mathar, Aug 31 2025

A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

Like all m-gons with m equal to a power of 2 (see A003401 and A000079), this is a constructible number.

			0.390180644032256535696569736954044481855383235503909615509004...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A000079, A003401.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/16], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/16)
```

Equals 2*sin(Pi/m) for m=16, 2*A232738. Equals also sqrt(2-sqrt(2+sqrt(2))).

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

A343060 Decimal expansion of tan(Pi/16).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.19891236737965800691159762264...

Cf. A232738 (sin(Pi/16)), A232737 (cos(Pi/16)), A343057 (tan(Pi/32)).

RealDigits[Tan[Pi/16], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/16)
```

sqrt((2-sqrt(2+sqrt(2)))/(2+sqrt(2+sqrt(2))))

PARI
```
sqrt(4+2*sqrt(2))-sqrt(2)-1
```

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

Equals sqrt(4+2*sqrt(2))-sqrt(2)-1.

cp's OEIS Frontend

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

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A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

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A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

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

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A343060 Decimal expansion of tan(Pi/16).

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