A073052 -id:A073052 - cp's OEIS frontend

A160389 Decimal expansion of 2*cos(Pi/7).

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

Offset: 1

Harry J. Smith, May 31 2009

Arises in the approximation of 14-fold quasipatterns by 14 Fourier modes.
Let DTS(n^c) denote the set of languages accepted by a deterministic Turing machine with space n^(o(1)) and time n^(c+o(1)), and let SAT denote the Boolean satisfiability problem. Then (1) SAT is not in DTS(n^c) for any c < 2*cos(Pi/7), and (2) the Williams inference rules cannot prove that SAT is not in DTS(n^c) for any c >= 2*cos(Pi/7). These results also apply to the Boolean satisfiability problem mod m where m is in A085971 except possibly for one prime. - Charles R Greathouse IV, Jul 19 2012
rho(7):= 2*cos(Pi/7) is the length ratio (smallest diagonal)/side in the regular 7-gon (heptagon). The algebraic number field Q(rho(7)) of degree 3 is fundamental for the 7-gon. See A187360 for the minimal polynomial C(7, x) of rho(7). The other (larger) diagonal/side ratio in the heptagon is sigma(7) = -1 + rho(7)^2, approx. 2.2469796. (see the decimal expansion in A231187). sigma(7) is the limit of a(n+1)/a(n) for n->infinity for the sequences like A006054 and A077998 which can be considered as analogs of the Fibonacci sequence in the pentagon. Thus sigma(7) plays in the heptagon the role of the golden section in the pentagon. See the P. Steinbach reference. - Wolfdieter Lang, Nov 21 2013
An algebraic integer of degree 3 with minimal polynomial x^3 - x^2 - 2x + 1. - Charles R Greathouse IV, Nov 12 2014
The other two solutions of the minimal polynomial of rho(7) = 2*cos(Pi/7) are 2*cos(3*Pi/7) and 2*cos(5*Pi/7). See eq. (20) of the W. Lang link. - Wolfdieter Lang, Feb 11 2015
The constant is the square root of 3.24697... (cf. A116425). It is the fifth-longest diagonal in the regular 14-gon with unit radius, which equals 2*sin(5*Pi/14). - Gary W. Adamson, Feb 14 2022

			1.801937735804838252472204639014890102331838324263714300107124846398864...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in non-integer bases, arXiv:1711.10397 [math.DS], 2017. See Theorem 1.1 p. 3.
Sam Buss and Ryan Williams, Limits on alternation-trading proofs for time-space lower bounds, Electronic Colloquium on Computational Complexity 2011
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], Oct 03 2012.
Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
Ryan Williams, Time-space tradeoffs for counting NP solutions modulo integers, Computational Complexity 17 (2008), pp. 179-219.
Index entries for algebraic numbers, degree 3

Cf. A039921 (continued fraction).
Cf. A047336, A047341, A116425, A255240, A255249.
Cf. A003558 (the constant is cyclic with period 3, for N = 7).

R:= RealField(200); Reverse(Intseq(Floor(10^110*2*Cos(Pi(R)/7)))); // Marius A. Burtea, Nov 13 2019

evalf(2*cos(Pi/7), 100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[2 Cos[Pi/7], 10, 111][[1]] (* Robert G. Wilson v, Jun 11 2013 *)

default(realprecision, 20080); x=2*cos(Pi/7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b160389.txt", n, " ", d));

Equals 2*A073052. - Michel Marcus, Nov 21 2013
Equals (Re((-(4*7)*(1 + 3*sqrt(3)*i))^(1/3)) + 1)/3, with the real part Re, and i = sqrt(-1). - Wolfdieter Lang, Feb 24 2015
Equals i^(2/7) - i^(12/7). - Peter Luschny, Apr 04 2020
From Peter Bala, Oct 20 2021: (Start)
Equals 2 - (1 - z)*(1 - z^6)/((1 - z^3)*(1 - z^4)), where z = exp(2*Pi*i/7).
The other two zeros of the minimal polynomial x^3 - x^2 - 2*x + 1 of 2*cos(Pi/7) are given by 2 - (1 - z^3)*(1 - z^4)/((1 - z^2)*(1 - z^5)) = 2*cos(3*Pi/7) = A255241 and 2 - (1 - z^2)*(1 - z^5)/((1 - z)*(1 - z^6)) = cos(5*Pi/7) = -A362922.
Equals Product_{n >= 0} (7*n+2)*(7*n+5)/((7*n+1)*(7*n+6)) = 1 + Product_{n >= 0} (7*n+2)*(7*n+5)/((7*n+3)*(7*n+4)) = 1/A255240.
The linear fractional mapping r -> 1/(1 - r) cyclically permutes the three zeros of the minimal polynomial x^3 - x^2 - 2*x + 1. The inverse mapping is r -> (r - 1)/r.
The quadratic mapping r -> 2 - r^2 also cyclically permutes the three zeros. The inverse mapping is r -> r^2 - r - 1. (End)
Equals i^(2/7) + i^(-2/7). - Gary W. Adamson, Feb 11 2022
From Amiram Eldar, Nov 22 2024: (Start)
Equals Product_{k>=1} (1 - (-1)^k/A047336(k)).
Equals 1 + cosec(3*Pi/14)/2 = 1 + Product_{k>=1} (1 + (-1)^k/A047341(k)). (End)
Equals sqrt(A116425). - Hugo Pfoertner, Nov 22 2024

A144981 Decimal expansion of cos(Pi/8) = cos(22.5 degrees).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 28 2008

Also the real part of i^(1/4). - Stanislav Sykora, Apr 25 2012
Width of a regular octagon of unit diameter. See Bingane and Audet. - Michel Marcus, Oct 04 2021
Minimal polynomial 8x^4 - 8x^2 + 1. - Charles R Greathouse IV, Oct 30 2023
Also the ratio (1+sqrt(2))/sqrt(4+2*sqrt(2)) of the radii and perimeters of the inscribed and circumscribed circles of a regular octagon. This and the first two comments are actually all equivalent. - M. F. Hasler, Aug 13 2025

			Equals 0.923879532511286756128183189396788286822416625863642486115097...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Christian Bingane and Charles Audet, The equilateral small octagon of maximal width, arXiv:2110.00036 [math.MG], 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Octagon.
Index entries for algebraic numbers, degree 4

Cf. A019863: cos(Pi/5), A010527: cos(Pi/6), A073052: cos(Pi/7), A019879: cos(Pi/9).

Maple
```
evalf(sqrt(2+sqrt(2))/2) ;
```

RealDigits[ Sqrt[2 + Sqrt[2]]/2, 10, 111][[1]] (* Or *) RealDigits[ Cos[Pi/8], 10, 111][[1]] (* Robert G. Wilson v *)

cos(Pi/8) \\ Michel Marcus, Dec 15 2015

from math import isqrt # integer arithmetic, avoiding 10^(4N) in inner isqrt
def A144981_first(N=99): return [9] if N<2 else list(map(int,str(
    isqrt(isqrt(100**(N+2)>>3)*10**(N-2)+100**N//2)))) # M. F. Hasler, Aug 13 2025

numerical_approx(sqrt(2+sqrt(2))/2, digits=120) # G. C. Greubel, Sep 04 2022

Equals sqrt(2 + sqrt(2))/2 = sqrt(3.41421...)/2 = 1.8477759.../2.
Equals Hypergeometric2F1([11/16, 5/16], [1/2], 3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2F1(-1/4,1/4;1/2;1/2) . - R. J. Mathar, Aug 31 2025

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

Original entry on oeis.org

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))).

A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232736.

Root of the equation -7 + 56*x^2 - 112*x^4 + 64*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.974927912181823607018131682993931217232785800619997437648...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
A. Arman, A. Bondarenko, and A. Prymak, Convex bodies of constant width with exponential illumination number, arXiv:2304.10418 [math.MG], 2023.
Index entries for algebraic numbers, degree 6

Cf. A232736 (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))), A232737 (real(I^(1/8))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

R:= RealField(100); Cos(Pi(R)/14); // G. C. Greubel, Sep 19 2022

RealDigits[Cos[Pi/14],10,120][[1]] (* Harvey P. Dale, Dec 15 2018 *)

numerical_approx(cos(pi/14), digits=120) # G. C. Greubel, Sep 19 2022

2*this^2 -1 = A073052. - R. J. Mathar, Aug 29 2025

Equals 2F1(-1/14,1/14;1/2;1) . - R. J. Mathar, Aug 31 2025

A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 26 2024

			1.0382606982861682835817694307429201653528601033...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 6.

Cf. A374957 (circumradius), A374972 (sagitta), A178817 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A178818, A073052, A343058.

Mathematica
```
First[RealDigits[Cot[Pi/7]/2, 10, 100]]
```

 5/(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/7)/2 = A178818/2.
Equals 1/(2*tan(Pi/7)) = 1/(2*A343058).
Equals A374957*cos(Pi/7) = A374957*A073052.
Equals A374957 - A374972.
Largest of the 6 real-valued roots of 448*x^6 -560*x^4 +84*x^2 -1 =0. - R. J. Mathar, Aug 29 2025

A323601 Decimal expansion of sin(Pi/7).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 19 2019

			0.43388373911755812047576833284835875460999072778745987644454730353220325...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A019829 (sin(Pi/9)), A232736 (sin(Pi/14)).

Cf. A121598, A272487.

SetDefaultRealField(RealField(100)); R:= RealField(); Sin(Pi(R)/7); // G. C. Greubel, Feb 08 2019

Mathematica
```
RealDigits[Sin[Pi/7], 10, 120][[1]]
```

default(realprecision, 100); sin(Pi/7) \\ G. C. Greubel, Feb 08 2019

polrootsreal(64*x^6-112*x^4+56*x^2-7)[4] \\ Charles R Greathouse IV, Feb 05 2025

numerical_approx(sin(pi/7), digits=100) # G. C. Greubel, Feb 08 2019

Root of the equation 64*x^6 - 112*x^4 + 56*x^2 - 7 = 0. (Other +- A232735 and +- 0.7818314... = +- cos(3*Pi/14))
Equals sqrt((196 + 7*i*2^(2/3)*(21*i*sqrt(3) - 7)^(1/3)*(i + sqrt(3)) + i*2^(4/3)*(21*i*sqrt(3) - 7)^(2/3)*(2*i + sqrt(3)))/336), where i is the imaginary unit.
Equals cos(5*Pi/14).
From Gleb Koloskov, Jul 15 2021: (Start)
Positive root of the equation x^3 + sqrt(7)/2*x^2 - sqrt(7)/8 = 0.
Equals ((4*sqrt(7)*(13+3*sqrt(3)*i))^(1/3)+28*(4*sqrt(7)*(13+3*sqrt(3)*i))^(-1/3)-2*sqrt(7))/12, where i is the imaginary unit. (End)
Equals 1/A121598 = A272487/2. - Hugo Pfoertner, Dec 15 2024
This^2 + A073052^2=1. - R. J. Mathar, Aug 31 2025

A343058 Decimal expansion of tan(Pi/7).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

Root of the equation -7 + 35*x^2 - 21*x^4 + x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.48157461880752864433216235305697...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A323601 (sin(Pi/7)), A073052 (cos(Pi/7)).

RealDigits[Tan[Pi/7], 10, 120][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/7)
```

numerical_approx(tan(pi/7), digits=123) # G. C. Greubel, Sep 30 2022

A374957 Decimal expansion of the circumradius of a regular heptagon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 26 2024

			1.15238243548124325262057511177342755672225094380...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 6.

Cf. A374971 (apothem), A374972 (sagitta), A178817 (area).
Cf. circumradius of other polygons with unit side length: A020760 (triangle), A010503 (square), A300074 (pentagon), A285871 (octagon), A375151 (9-gon), A001622 (10-gon), A375190 (11-gon), A188887 (12-gon).
Cf. A073052, A121598, A272487.

Mathematica
```
First[RealDigits[Csc[Pi/7]/2, 10, 100]]
```

 5/sin(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals csc(Pi/7)/2 = A121598/2.
Equals 1/(2*sin(Pi/7)) = 1/A272487.
Equals A374971/cos(Pi/7) = A374971/A073052.
Largest of the 6 real-valued roots of 7*x^6-14*x^4+7*x^2-1=0. - R. J. Mathar, Aug 29 2025

A362922 Decimal expansion of cos(2Pi/7) = sin(3Pi/14) = A255249/2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 25 2023

This number, negated, is a zero of the polynomial 8*x^3 - 4*x^2 - 4*x + 1 that arises in the dissection of a regular heptagon. The other two zeros are cos(Pi/7) (A073052) and sin(Pi/14) (A232736).

The old definition was: Decimal expansion of 1/(8*cos(Pi/7)*sin(Pi/14)).

			0.6234898018587335305250048840042398106322747308964021053655...

Cf. A073052, A160389, A232736, A255249.

Digits := 110: evalf(((-1)^(2/7) - (-1)^(5/7))/2, Digits)*10^96:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Jun 25 2023

First@ RealDigits[1/(8*Cos[Pi/7]*Sin[Pi/14]), 10, 96] (* Michael De Vlieger, Jun 25 2023 *)

Equals 1/(4*cos(Pi/7)-2) = 1/(2*A160389-2). - Alois P. Heinz, Jun 25 2023

Equals (i^(4/7) - i^(10/7))/2. - Peter Luschny, Jun 26 2023

Simpler definition from Alois P. Heinz, Jun 25 2023.

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

A160389 Decimal expansion of 2*cos(Pi/7).

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A144981 Decimal expansion of cos(Pi/8) = cos(22.5 degrees).

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

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A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

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A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

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A323601 Decimal expansion of sin(Pi/7).

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A362922 Decimal expansion of cos(2Pi/7) = sin(3Pi/14) = A255249/2.