A010527 -id:A010527 - cp's OEIS frontend

A019842 Decimal expansion of sine of 33 degrees.

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

Offset: 0

N. J. A. Sloane

This sequence is also decimal expansion of cosine of 57 degrees. - Mohammad K. Azarian, Jun 29 2013

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

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

RealDigits[Sin[33 Degree],10,120][[1]] (* Harvey P. Dale, Jul 26 2023 *)

sin(11*Pi/60) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019831 * A019888 + A019820 * A019877 = (1/2) * A019896 + A019812 * A010527. - R. J. Mathar, Jan 27 2021

A019844 Decimal expansion of sine of 35 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This sequence is also decimal expansion of cosine of 55 degrees. - Mohammad K. Azarian, Jun 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

RealDigits[Sin[35 Degree],10,120][[1]] (* Harvey P. Dale, Mar 26 2013 *)

sin(35*Pi/180) \\ Michel Marcus, Aug 11 2014

Equals A019837 * A019892 + A019816 * A019871 = (1/2)* A019894 + A019814 * A010527. - R. J. Mathar, Jan 27 2021

A041132 Numerators of continued fraction convergents to sqrt(75).

Original entry on oeis.org

8, 9, 17, 26, 433, 459, 892, 1351, 22508, 23859, 46367, 70226, 1169983, 1240209, 2410192, 3650401, 60816608, 64467009, 125283617, 189750626, 3161293633, 3351044259, 6512337892, 9863382151, 164326452308, 174189834459, 338516286767, 512706121226, 8541814226383

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,52,0,0,0,-1).

Cf. A010527, A041133.

Numerator[Convergents[Sqrt[75], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

G.f.: -(x^7-8*x^6+9*x^5-17*x^4-26*x^3-17*x^2-9*x-8) / (x^8-52*x^4+1). - Colin Barker, Nov 05 2013

More terms from Colin Barker, Nov 05 2013

A093604 Decimal expansion of D/2, where D^2 = 3*sqrt(3)/Pi.

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, May 14 2004

D/2=sqrt(3*sqrt(3)/Pi)/2 corresponds to the radius of the area-bisecting concentric circle within the unit-sided hexagon.

			sqrt(3*sqrt(3)/Pi)/2 = 0.6430370685787437846417825056651579788623049833326304871239...

Index entries for transcendental numbers

Cf. A097603, A010527, A011002, A087197. - R. J. Mathar, Feb 06 2009

RealDigits[Sqrt[(3Sqrt[3])/Pi]/2,10,120][[1]] (* Harvey P. Dale, Aug 27 2017 *)

sqrt(sqrt(27)/Pi)/2 \\ Charles R Greathouse IV, Oct 01 2022

Removed leading zero and adjusted offset - R. J. Mathar, Feb 06 2009

Corrected and extended by Harvey P. Dale, Aug 27 2017

A220351 Decimal expansion of (3*sqrt(3)+sqrt(7))/10.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Dec 11 2012

Smith & Smith conjecture that this is the Steiner ratio rho_3, the least upper bound on the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 3. Diaconis & Graham offer $1000 for proof (or disproof) of this conjecture.

This is an algebraic number of degree 4; the minimal polynomial is 25x^4 - 17x^2 + 1.

			0.7841903733771222471083954778156877526538674944513592064535755...

Persi Diaconis and R. L. Graham, Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks, Princeton University Press, 2011. See pp. 212-214.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.6 Steiner Tree Constants, p. 504.

Warren D. Smith and J. MacGregor Smith, On the Steiner ratio in 3-space, Journal of Combinatorial Theory, Series A 69:2 (1995), pp. 301-332.
Index entries for algebraic numbers, degree 4

Cf. A010527.

RealDigits[(3*Sqrt[3]+Sqrt[7])/10, 10, 120] // First (* Jean-François Alcover, May 27 2014 *)

PARI
```
(3*sqrt(3)+sqrt(7))/10
```

polrootsreal(25*x^4 - 17*x^2 + 1)[4] \\ Charles R Greathouse IV, Jan 05 2016

(3*sqrt(3)+sqrt(7))/10.

Formula and name simplified by Jean-François Alcover, May 27 2014

A277754 Decimal expansion of Sum_{n>=1} sin((n*Pi)/3)^n.

Original entry on oeis.org

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

Offset: 1

Michelle Huff, Oct 28 2016

The sum of four easily recognized geometric series that have common ratio 9/64.

An algebraic number with minimal polynomial 1369x^2 - 6216x + 6468. - Charles R Greathouse IV, Oct 29 2016

			2.92564084610714275971308780...

Michelle Huff, Table of n, a(n) for n = 1..10000

Cf. A010527, A277755.

r = Sqrt[3]/2; u = Simplify[r (1 + r + r^3 - r^4)/(1 - r^6)]
RealDigits[N[u, 120], 10][[1]]

suminf(n=1, sin((n*Pi)/3)^n) \\ Michel Marcus, Oct 29 2016

(sqrt(3)+6)*14/37 \\ Charles R Greathouse IV, Oct 29 2016

Equals (14/37)*(6 + sqrt(3)).

A277755 Decimal expansion of Sum_{n>=1} |sin((n*Pi)/3)|^n.

Original entry on oeis.org

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

Offset: 1

Michelle Huff, Oct 28 2016

			(2/37)*(42 + 25 Sqrt[3]) = 4.61087946968767201828033289392685454992..., the sum of two easily recognized geometric series that have common ratio (3/4)^(3/2).

Michelle Huff, Table of n, a(n) for n = 1..10000

Cf. A010527, A277754.

r = Sqrt[3]/2; u = Simplify[r (1 + r)/(1 - r^3)]
RealDigits[N[u, 1200], 10][[1]]

suminf(n=1, abs(sin((n*Pi)/3))^n) \\ Michel Marcus, Oct 29 2016

A325732 First term of n-th difference sequence of (floor(k*r)), r = sqrt(3/4), k >= 0.

Original entry on oeis.org

0, 1, -1, 1, -1, 1, -1, 0, 7, -35, 119, -329, 791, -1715, 3430, -6419, 11319, -18767, 28763, -38759, 38759, 1, -149228, 572057, -1615429, 3979001, -9014851, 19251001, -39309301, 77558760, -149239771, 282712561, -532577025, 1008032953, -1934671809, 3787949521

Offset: 1

Clark Kimberling, May 20 2019

Clark Kimberling, Table of n, a(n) for n = 1..200

Cf. A325664. Inverse binomial Transform of A171970 and A172475.

Cf. A010527 (sqrt(3/4)).

Table[First[Differences[Table[Floor[Sqrt[3/4]*n], {n, 0, 50}], n]], {n, 1, 50}]

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

A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.9951847266721968862448369...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 16

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Cos[Pi/32], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
real(I^(1/16))
```
PARI
```
cos(Pi/32)
```
PARI
```
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).
Satisfies 32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +84480*x^8 -21504*x^6 +2688*x^4 -128*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

cp's OEIS Frontend

A019842 Decimal expansion of sine of 33 degrees.

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A019844 Decimal expansion of sine of 35 degrees.

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A041132 Numerators of continued fraction convergents to sqrt(75).

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A093604 Decimal expansion of D/2, where D^2 = 3*sqrt(3)/Pi.

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A220351 Decimal expansion of (3*sqrt(3)+sqrt(7))/10.

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A277754 Decimal expansion of Sum_{n>=1} sin((n*Pi)/3)^n.

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A277755 Decimal expansion of Sum_{n>=1} |sin((n*Pi)/3)|^n.

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