A019815 -id:A019815 - cp's OEIS frontend

A019896 Decimal expansion of sine of 87 degrees.

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

Offset: 0

N. J. A. Sloane

Equals sin(29*Pi/60). - Wesley Ivan Hurt, Sep 01 2014

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.998629534754573873784492058439436580590952290767785532441441...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Raimundas Vidunas, Expressions for values of the gamma function, math.CA/0403510
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

Digits:=100: evalf(sin(29*Pi/60)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sin[87Degree], 10, 100][[1]] (* Alonso del Arte, Aug 31 2014 *)

cos(Pi/60) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/60) = [5-sqrt(5)]*[1+sqrt(3)]*[2-sqrt(3)+sqrt{5+2*sqrt(5)}]/[8*sqrt(10)] = sqrt[(1+A019893)/2]. - R. J. Mathar, Jun 18 2006

Equals A019867 * A019870 + A019838 * A019841 = A019893 * A019896 + A019812 * A019815. - R. J. Mathar, Jan 27 2021

A019818 Decimal expansion of sine of 9 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the imaginary part of i^(1/10). - Stanislav Sykora, Apr 25 2012

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			sin(Pi/20) = 0.1564344650402308690101053194671668923138...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

Cf. A019812, A019815, A019893, A019896.

RealDigits[Sin[9 Degree],10,120][[1]] (* Harvey P. Dale, Sep 15 2014 *)

sin(Pi/20) \\ Charles R Greathouse IV, Aug 27 2017

sqrt(8-2*sqrt(10+2*sqrt(5)))/4 = sqrt((1/2)*(1 - sqrt((1/8)*(5 + sqrt(5))))). - Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 13 2006
Equals A019815*A019896 + A019812*A019893. - R. J. Mathar, Jan 27 2021
This^2 + A019890^2=1. - R. J. Mathar, Aug 31 2025
Smallest positive of the 8 real-valued roots of 256*x^8 -512*x^6 +304*x^4 -48*x^2+1=0. - R. J. Mathar, Aug 31 2025

A019821 Decimal expansion of sine of 12 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.20791169...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

Cf. A019815, A019887, A019893.

RealDigits[N[Sin[Pi/15], 110]][[1]] (* Wesley Ivan Hurt, Dec 27 2021 *)

sin(Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals sin(Pi/15) = sqrt(1-A019887^2) = (sqrt(5)-1)*(sqrt(5+2*sqrt(5)) - sqrt(3))/8. - R. J. Mathar, Jun 18 2006
Equals 2*A019815*A019893. - R. J. Mathar, Jan 17 2021
Smallest positive of the 8 real-valued roots of 256*x^8-448*x^6+224*x^4-32*x^2+1 =0. (Other A019893, A019833, A019857)- R. J. Mathar, Aug 31 2025

A019816 Decimal expansion of sine of 7 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 48 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 48

RealDigits[Sin[7 Degree],10,120][[1]] (* Harvey P. Dale, Mar 25 2016 *)

sin(7*Pi/180) \\ Charles R Greathouse IV, Aug 27 2017

Also cosine of 83 degrees. - Charles R Greathouse IV, Aug 27 2017

Equals A019815 * A019898 + A019810 * A019893. - R. J. Mathar, Jan 27 2021

A306603 a(n) = (2 cos(Pi/15))^n + (2 cos(7 Pi/15))^n + (2 cos(11 Pi/15))^n + (2 cos(13 Pi/15))^n.

Original entry on oeis.org

4, -1, 9, -1, 29, 4, 99, 34, 349, 179, 1254, 824, 4559, 3574, 16704, 15004, 61549, 61709, 227799, 250229, 846254, 1004149, 3153984, 3997399, 11788879, 15812504, 44178624, 62229509, 165946124, 243873904, 624650004, 952400599, 2355748909, 3708579599

Offset: 0

Greg Dresden, Feb 27 2019

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = -1, e_2 = -4, e_3 = -4 and e_4 = 1. The arguments are e_j(x_1, x_2, x_3, x_4), for j = 1..4, with the zeros {x_i}A187360,%20for%20n%20=%2015),%20appearing%20to%20the%20power%20n%20in%20the%20formula%20given%20above.%20-%20_Wolfdieter%20Lang">{i=1..4} of the minimal polynomial of 2*cos(Pi/15) (see A187360, for n = 15), appearing to the power n in the formula given above. - _Wolfdieter Lang, May 08 2019

Index entries for linear recurrences with constant coefficients, signature (-1,4,4,-1).

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A324602.

Table[Sum[(2.0 Cos[k Pi/15])^n, {k, {1, 7, 11, 13}}] // Round, {n, 1, 30}]
LinearRecurrence[{-1,4,4,-1},{4,-1,9,-1},40] (* Harvey P. Dale, Jun 02 2024 *)

G.f.: (4*x^3+8*x^2-3*x-4)/(-x^4+4*x^3+4*x^2-x-1). - Alois P. Heinz, Feb 27 2019

a(n) = -a(n-1) + 4*a(n-2) + 4*a(n-3) -a(n-4). - Greg Dresden, Feb 27 2019

A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.

Original entry on oeis.org

4, 24, 109, 524, 2504, 11979, 57299, 274084, 1311049, 6271254, 29997829, 143491199, 686373809, 3283190949, 15704770004, 75121978804, 359337430474, 1718849676159, 8221921677724, 39328626006254, 188124003629279, 899869747188249, 4304424455586134

Offset: 1

Greg Dresden, Feb 28 2019

-a(n) is the coefficient of x in the minimal polynomial for (2*cos(Pi/15))^n, for n >= 1. The coefficients of -x^3 are A306603(n), and those of x^2 are A306611(n).

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = 4, e_2 = -4, e_3 = -1 and e_4 = 1. The arguments are e_j(1/x_1, 1/x_2, 1/x_3, 1/x_4), for j = 1..4, with the zeros {x_i}{i=1..4} of the minimal polynomial of 2*cos(Pi/15), appearing under the negative powers of the formula given above. - _Wolfdieter Lang, May 08 2019

Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1).

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A306603 (positive powers of these cosines), A306611, A324602.

Table[Round[N[Sum[(2 Cos[k Pi/15])^(-n), {k,{1,7,11,13}}],50]],{n,1,30}]

a(n) = 4a(n-1) + 4a(n-2) - a(n-3) - a(n-4).
G.f.: x*(-4x^3 -3x^2 +8x +4)/(x^4 +x^3 -4x^2 -4x +1).
a(n) = round((2*cos(7*Pi/15))^(-n)) for n >= 3.

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

A371983 Decimal expansion of Gamma(1/30).

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Apr 15 2024

			29.4547797456996940196962082886383457347018736055729711046565415567498...

Albert Nijenhuis, Small Gamma Products with Simple Values, arXiv:0907.1689v1 [math.CA], 2009.
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
Index to sequences related to gamma function

Cf. A073005, A175380, A256191, A203140, A371881.

evalf(GAMMA(1/30), 130);  # Alois P. Heinz, Apr 15 2024

RealDigits[Gamma[1/30], 10, 120][[1]]
RealDigits[2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma[1/5] * Gamma[1/3] / ((10 + Sqrt[5] - Sqrt[75 + 30*Sqrt[5]])^(1/4) * Sqrt[Pi]), 10, 120][[1]]

Equals 3^(9/20) * sqrt(5 + sqrt(5)) * sqrt(sqrt(15) + sqrt(5 + 2*sqrt(5))) * Gamma(1/3) * Gamma(1/5) / (sqrt(Pi) * 2^(16/15) * 5^(1/6)).
Equals 2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma(1/5) * Gamma(1/3) / ((10 + sqrt(5) - sqrt(75 + 30*sqrt(5)))^(1/4) * sqrt(Pi)).
Equals 8*Pi^2 / (Gamma(17/30) * Gamma(19/30) * Gamma(23/30)).
Equals Gamma(7/30) * Gamma(11/30) * Gamma(13/30) / (2*Pi*A019815).

A307886 Array of coefficients of the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 (ascending powers).

Original entry on oeis.org

1, -4, -4, 1, 1, 1, -24, 26, -9, 1, 1, -109, -49, 1, 1, 1, -524, 246, -29, 1, 1, -2504, -619, -4, 1, 1, -11979, 2621, -99, 1, 1, -57299, -7774, -34, 1, 1, -274084, 30126, -349, 1, 1, -1311049, -97879, -179, 1, 1, -6271254, 363131, -1254, 1, 1, -29997829, -1237504, -824, 1

Offset: 1

Greg Dresden and Wolfdieter Lang, May 02 2019

The length of each row is 5.

The minimal polynomial of (2*cos(Pi/15))^n, for n >= 1, is C(15, n, x) = Product_{j=0..3} (x - (x_j)^n) = Sum_{k=0} T(n, k) x^k, where x_0 = 2*cos(Pi/15), x_1 = 2*cos(7*Pi/15), x_2 = 2*cos(11*Pi/15), and x_3 = 2*cos(13*Pi/15) are the zeros of C(15, 1, x) with coefficients given in A187360 (row n=15).

			The rectangular array T(n, k) begins:
n\k 0      1      2      3      4
---------------------------------
1:  1     -4     -4      1      1
2:  1    -24     26     -9      1
3:  1   -109    -49      1      1
4:  1   -524    246    -29      1
5:  1  -2504   -619     -4      1
6:  1 -11979   2621    -99      1
7:  1 -57299  -7774    -34      1
...

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A306603, A306610, A306611.

Flatten[Table[CoefficientList[MinimalPolynomial[(2*Cos[\[Pi]/15])^n, x], x], {n, 1, 15}]]

T(n,k) = the coefficient of x^k in C(15, n, x), n >= 1, k=0,1,2,3,4, with C(15, n, k) the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 as defined above.

T(n, 0) = T(n, 4) = 1. T(n, 1) = -A306610(n), T(n, 2) = A306611(n), T(n, 3) = -A306603(n), n >= 1.

cp's OEIS Frontend

A019896 Decimal expansion of sine of 87 degrees.

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A019818 Decimal expansion of sine of 9 degrees.

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A019821 Decimal expansion of sine of 12 degrees.

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A019816 Decimal expansion of sine of 7 degrees.

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A306603 a(n) = (2 cos(Pi/15))^n + (2 cos(7 Pi/15))^n + (2 cos(11 Pi/15))^n + (2 cos(13 Pi/15))^n.

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A306610 a(n) = (2*cos(Pi/15))^(-n) + (2*cos(7*Pi/15))^(-n) + (2*cos(11*Pi/15))^(-n) + (2*cos(13*Pi/15))^(-n), for n >= 1.

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

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A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.