A019827 -id:A019827 - cp's OEIS frontend

A019916 Decimal expansion of tan(Pi/10) (angle of 18 degrees).

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

Offset: 0

N. J. A. Sloane

In a regular pentagon inscribed in a unit circle this is the cube of the length of the side divided by 5: (1/5)*(sqrt(3 - phi))^3 with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Quartic number of denominator 5 and minimal polynomial 5x^4 - 10x^2 + 1. - Charles R Greathouse IV, May 13 2019
The other positive root of the minimal polynomial is A019952. - R. J. Mathar, Sep 06 2025

			0.3249196962329063261558714122151344649549034715214751003078047191...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827 (sin(Pi/10)), A019881 (cos(Pi/10)).

RealDigits[Tan[18 Degree],10,120][[1]] (* Harvey P. Dale, Mar 07 2012 *)

tan(Pi/10) \\ Michel Marcus, Jan 08 2018

polrootsreal(5*x^4-10*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals A019827/A019881 = 1/A019970 = 1/sqrt(5+2*sqrt(5)). - R. J. Mathar, Jul 26 2010
Equals tan((phi - 1)/sqrt(2 + phi)) = (1/5)*(sqrt(3 - phi))^3 = (3 - phi)*sqrt(3 - phi)/5 = sqrt(7 - 4*phi)/(2*phi - 1), with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Equals Product_{k>=0} ((5*k + 1)/(5*k + 4))^(-1)^(k) = Product_{k>=0} A090771(k)/A090773(k). - Antonio Graciá Llorente, Mar 24 2024
Equals A019845/(1+A019863). - R. J. Mathar, Sep 06 2025

A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jul 01 2014

Essentially the same digit sequence as A239798, A019863 and A019827. - R. J. Mathar, Jul 03 2014

The minimal polynomial of this constant is x^2 - 11*x - 1. - Joerg Arndt, Jan 01 2017

			11.09016994374947424102293417182819058860154589902881431067724311352630...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 83.

Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
Wikipedia, Hard Hexagon Model.
D. W. Wood and R. W. Turnbull, z^2-11z-1 as an algebraic invariant for the hard-hexagon model, 1988 J. Phys. A: Math. Gen. 21 L989.

Cf. A019863, A019827, A085850, A239798.

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458 A176522, A176537 (metallic means).

RealDigits[GoldenRatio^5, 10, 103] // First

(5*sqrt(5)+11)/2 \\ Charles R Greathouse IV, Aug 10 2016

Equals ((1 + sqrt(5))/2)^5 = (11 + 5*sqrt(5))/2.
Equals phi^5 = 11 + 1/phi^5 = 3 + 5*phi, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Nov 11 2023
Equals lim_{n->infinity} S(n, 5*(-1 + 2*phi))/ S(n-1, 5*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A280533 Decimal expansion of 14*sin(Pi/14).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 04 2017

Decimal expansion of the ratio of the perimeter of a regular 7-gon (heptagon) to its diameter (largest diagonal).

			3.115293075388401660044635902955126632528977962703629374367818222563897248...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280585 (n=8), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A232736.

RealDigits[14*Sin[Pi/14], 10, 129][[1]] (* G. C. Greubel, Sep 20 2022 *)

PARI
```
14*sin(Pi/14)
```

numerical_approx(14*sin(pi/14), digits=127) # G. C. Greubel, Sep 20 2022

Equals 14*A232736.

A019836 Decimal expansion of sine of 27 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 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 8

RealDigits[Sin[27 Degree],10,120][[1]] (* Harvey P. Dale, Nov 09 2014 *)

cos(7*Pi/20) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019827 * A019890 + A019818 * A019881 = A019833 * A019896 + A019812 * A019875. - R. J. Mathar, Jan 27 2021

Equals cos(7*Pi/20). 2*this^2-1 = -A019845. - R. J. Mathar, Aug 29 2025

A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232737.

			0.195090322016128267848284868477022240927691617751954807754502...

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. A232737 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232736 (imag(I^(1/7))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

R:= RealField(116); Sin(Pi(R)/16); // G. C. Greubel, Sep 20 2022

RealDigits[Sin[Pi/16],10,120][[1]] (* Harvey P. Dale, Sep 01 2018 *)

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

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

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

numerical_approx(sin(pi/16), digits=116) # G. C. Greubel, Sep 20 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
This^2 + A232737^2 = 1.
Smallest positive of the 8 real-valued roots of 128*x^8-256*x^6+160*x^4-32*x^2+1=0.
Equals A182168/(2*A232737). - R. J. Mathar, Sep 05 2025

A280725 Decimal expansion of 22*sin(Pi/22).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 07 2017

The ratio of the perimeter of a regular 11-gon (hendecagon) to its diameter (largest diagonal).

Also least positive root of x^5 - 11x^4 - 484x^3 + 3993x^2 + 43923x - 161051.

			3.130926442012273089763438709560132713403129944771655225197821304298120771...

Index entries for algebraic numbers, degree 5.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280585 (n=8), A280633 (n=9), A280819 (n=12).

evalf(22*sin(Pi/22),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[22*Sin[Pi/22], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
22*sin(Pi/22)
```

A204188 Decimal expansion of sqrt(5)/4.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Jan 14 2012

Equals Product_{n>=1} (1 - 1/A000032(2^n)).
Essentially the same as A019863 and A019827. - R. J. Mathar, Jan 16 2012
The value is the distance of the W point of the Wigner-Seitz cell of the body-centered cubic lattice (that is the Brioullin zone of the face-centered cubic lattice) to its four nearest neighbors. Let the points of the simple cubic lattice be at (1,0,0), (0,1,0), (1,0,0) etc and the point in the cube center at (1/2, 1/2, 1/2). Then W is at (0, 1/4, 1/2) [or any of the 24 symmetry related positions like (0, 3/4, 1/2), (0, 1/2, 1/4) etc.], and the four lattice points closest to W are at (-1/2, 1/2, 1/2), (0,0,0), (1/2, 1/2, 1/2) and (0,0,1). - R. J. Mathar, Aug 19 2013

			0.5590169943749474241022934171828190588601545899028814310677243113526302...

J. Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
Y. Tachiya, Transcendence of certain infinite products, J. Number Theory 125 (2007), 182-200.
Wikipedia, Brillouin zone

Cf. A001622, A002163.

Maple
```
evalf(sqrt(5)/4);
```

RealDigits[Sqrt[5]/4, 10, 100][[1]] (* Amiram Eldar, Dec 04 2018 *)

sqrt(5)/4 \\ Charles R Greathouse IV, Apr 21 2016

Equals sqrt(5)/4 = (-1 + 2*phi)/4, with the golden section phi from A001622.

Equals 5*A020837.

A280585 Decimal expansion of 8*sin(Pi/8).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 05 2017

Decimal expansion of the ratio of the perimeter of a regular 8-gon (octagon) to its diameter (largest diagonal).

			3.061467458920718173827679872243190934090756499885016331470405085020368271...

Index entries for algebraic numbers, degree 4.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A182168.

evalf(8*sin(Pi/8),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[8*Sin[Pi/8], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
8*sin(Pi/8)
```

Equals 8*A182168.

A280633 Decimal expansion of 18*sin(Pi/18).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 06 2017

The ratio of the perimeter of a regular 9-gon (nonagon) to its diameter (largest diagonal).

Also least positive root of x^3 - 243x + 729.

			3.125667198004746279330899281847666328006762189313248970252344806377184798...

Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7),A280585 (n=8), A280725(n=11), A280819 (n=12).

evalf(18*sin(Pi/18),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[18*Sin[Pi/18],10,120][[1]] (* Harvey P. Dale, Dec 02 2018 *)

PARI
```
18*sin(Pi/18)
```

18*A019819.

A280819 Decimal expansion of 12*sin(Pi/12).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 08 2017

The ratio of the perimeter of a regular 12-gon (dodecagon) to its diameter (greatest diagonal).

A quartic integer: the least positive root of x^4 - 144x^2 + 1296.

			3.105828541230249148186786051488579940188826815839166165768038487780683696...

Index entries for algebraic numbers, degree 4

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280585 (n=8), A280633 (n=9), A280725 (n=11).

First@ RealDigits@ N[12 Sin[Pi/12], 120] (* Michael De Vlieger, Feb 05 2017 *)

PARI
```
12*sin(Pi/12)
```

polrootsreal(x^4-144*x^2+1296)[3] \\ Charles R Greathouse IV, Feb 05 2017

12*A019824.

cp's OEIS Frontend

A019916 Decimal expansion of tan(Pi/10) (angle of 18 degrees).

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A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

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A280533 Decimal expansion of 14*sin(Pi/14).

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A019836 Decimal expansion of sine of 27 degrees.

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A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

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A280725 Decimal expansion of 22*sin(Pi/22).

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A204188 Decimal expansion of sqrt(5)/4.

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