A178182 -id:A178182 - cp's OEIS frontend

A093819 Algebraic degree of sin(2*Pi/n).

1, 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 1, 12, 6, 8, 4, 16, 6, 18, 2, 12, 10, 22, 4, 20, 12, 18, 3, 28, 8, 30, 8, 20, 16, 24, 3, 36, 18, 24, 8, 40, 12, 42, 5, 24, 22, 46, 8, 42, 20, 32, 6, 52, 18, 40, 12, 36, 28, 58, 4, 60, 30, 36, 16, 48, 20, 66, 8, 44, 24, 70, 12, 72, 36, 40, 9, 60, 24

Offset: 1

Eric W. Weisstein, Apr 16 2004

nonn

The degree formula given in the I. Niven reference on p. 37-8 (see below) appears as part of theorem 3.9 attributed to D. H. Lehmer. However, this part, concerning sin(2*Pi/n), differs from Lehmer's result, which in fact is incorrect. - Wolfdieter Lang, Jan 09 2011
This is also the algebraic degree of the area of a regular n-gon inscribed in the unit circle. - Jack W Grahl, Jan 10 2011
Every degree appears in this sequence except for the half-nontotients, A079695. - T. D. Noe, Jan 12 2011
See A181872/A181873 for the monic rational minimal polynomial of sin(2*Pi/n), and A181871 for the non-monic integer version. In A231188 the (monic and integer) minimal polynomials for 2*sin(2*Pi/n) are given. - Wolfdieter Lang, Nov 30 2013

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

T. D. Noe, Table of n, a(n) for n = 1..1000
Sameen Ahmed Khan, Trigonometric Ratios Using Algebraic Methods, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.
Eric Weisstein's World of Mathematics, Trigonometry Angles

Cf. A055035, A023022 (alg. degree of cos(2*Pi/n)), A183919.

Cf. A181872/A181873, A181871, A231188. - Wolfdieter Lang, Nov 30 2013

a[4]=1; a[n_] := Module[{g=GCD[n, 8], e=EulerPhi[n]}, If[g<4, e, If[g==4, e/4, e/2]]]; Array[a, 1000]
f[n_] := Exponent[ MinimalPolynomial[ Sin[ 2Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)

a(4)=1, a(n)=phi(n) if gcd(n,8)<4; a(n)=phi(n)/4 if gcd(n,8)=4, and a(n)=phi(n)/2 if gcd(n,8)>4. Here phi(n)=A000010(n) (Euler totient). See the I. Niven reference, Theorem 3.9, p. 37-8. - Wolfdieter Lang, Jan 09 2011

a(n) = delta(c(n)/2) if c(n) = A178182(n) is even, and delta(c(n)) if c(n) is odd, with delta(n) = A055034(n), the degree of the algebraic number 2*cos(Pi/n). - Wolfdieter Lang, Nov 30 2013

A181872 Numerators of coefficient array for minimal polynomials of sin(2*Pi/n). Rising powers of x.

Original entry on oeis.org

0, 1, 0, 1, -3, 0, 1, -1, 1, 5, 0, -5, 0, 1, -3, 0, 1, -7, 0, 7, 0, -7, 0, 1, -1, 0, 1, -3, 0, 9, 0, -3, 0, 1, 5, 0, -5, 0, 1, -11, 0, 55, 0, -77, 0, 11, 0, -11, 0, 1, -1, 1, 13, 0, -91, 0, 91, 0, -39, 0, 65, 0, -13, 0, 1, -7, 0, 7, 0, -7, 0, 1, 1, 0, -1, 0, 7, 0, -7, 0, 1, 1, 0, -1, 0, 1, 17, 0, -51, 0, 357, 0, -561, 0, 935, 0, -221, 0, 119, 0, -17, 0, 1, -3, 0, 9, 0, -3, 0, 1, -19, 0, 285, 0, -627, 0, 627, 0, -2717, 0, 1729, 0, -665, 0, 19, 0, -19, 0, 1, -1, 1, 1, 1, 0, -1, 0, 15, 0, -39, 0, 11, 0, -11, 0, 1, -11, 0, 55, 0, -77, 0, 11, 0, -11, 0, 1

Offset: 1

Wolfdieter Lang, Jan 13 2011

The corresponding denominator array is given in A181873(n,m).
The sequence of row lengths of this array is A093819(n)+1: [2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, ...].
The minimal polynomial of the algebraic number sin(2*Pi/n), n >= 1, is here called Pi(n,x) := Sum_{m=0..d(n)} r(n,m)*x^m with the degree sequence d(n):=A093819(n), and the rationals r(n):=a(n,m)/b(n,m) with b(n,m):=A181873(n,m).
  See the Niven reference, p. 28, for the definition of 'minimal polynomial of an algebraic number'.
  Minimal polynomials are irreducible.
  The minimal polynomials of sin(2*Pi/n) are treated, e.g., in the Lehmer, Niven and Watkins-Zeitlin references.
The minimal polynomials Pi(n,x) of sin(2*Pi/n) are found from Psi(c(n),x), where Psi(m,x) is the minimal polynomial of cos(2*Pi/m), and
  c(n):= denominator(|(4-n)/(4*n)|) = A178182(n).
For the regular n-gon inscribed in the unit circle the area is n*sin(2*Pi/n). See the remark by Jack W Grahl under A093819.
S. Beslin and V. de Angelis (see the reference) give an explicit formula for the (integer) minimal polynomial of sin(2*Pi/p), called S_p(x), and cos(2*Pi/p), called C_p(x),for odd prime p, p=2k+1, with the results:
  S_p(x) = Sum_{l=0..k} ((-1)^l)*binomial(p,2*l+1)*(1-x^2)^(k-l)*x^(2*l), and C_p(x) = S_p(sqrt((1-x)/2)), where S_p(x), with leading term ((-2)^k))*x^(p-1), checks with((-2)^k)*Pi(p,x). - Wolfdieter Lang, Feb 28 2011
The zeros of Pi(n, x) result from those of the minimal polynomial Psi(n, x) of cos(2*Pi/n), and they are cos(2*Pi*k/n), for k = 0, ..., floor(c(n)/2), with c(n) = A178182(n), and restriction gcd(k, c(n)) = 1, for n >= 1. There are d(n) = A093819(n) such zeros. - Wolfdieter Lang, Oct 30 2019

			Triangle begins:
  [0, 1],
  [0, 1],
  [-3, 0, 1],
  [-1, 1],
  [5, 0, -5, 0, 1],
  [-3, 0, 1],
  [-7, 0, 7, 0, -7, 0, 1],
  [-1, 0, 1],
  [-3, 0, 9, 0, -3, 0, 1],
  [5, 0, -5, 0, 1],
  ...
The rational coefficients r(n,m) start like:
  [0, 1],
  [0, 1],
  [-3/4, 0, 1],
  [-1, 1],
  [5/16, 0, -5/4, 0, 1],
  [-3/4, 0, 1],
  [-7/64, 0, 7/8, 0, -7/4, 0, 1],
  [-1/2, 0, 1],
  [-3/64, 0, 9/16, 0, -3/2, 0, 1],
  ...
Pi(6,n) = Psi(c(6),x) = Psi(12,x) = x^2-3/4.

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

S. Beslin and V. de Angelis, The minimal Polynomials of sin(2Pi/p) and cos(2Pi/p), Mathematics Mag. 77.2 (2004) 146-9.
Wolfdieter Lang, Minimal polynomials for sin(2Pi/n).
D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40 (3) (1933) 165-6.
W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.

Cf. A181875, A181876 (minimal polynomials of cos(2*Pi/n)).

p[n_, x_] := MinimalPolynomial[ Sin[2 Pi/n], x]; Flatten[ Numerator[ Table[ coes = CoefficientList[ p[n, x], x]; coes / Last[coes], {n, 1, 22}]]] (* Jean-François Alcover, Nov 07 2011 *)

a(n,m) = numerator([x^m]Pi(n,x)), n>=1, m=0..A093819(n). For Pi(n,x) see the comments.

The minimal polynomial Pi(n,x) = Product_{k=0..floor(c(n)/2), gcd(k, c(n)) = 1}, x - cos(2*Pi*k/c(n)), for n >= 1. - Wolfdieter Lang, Oct 30 2019

A225975 Square root of A226008(n).

Original entry on oeis.org

0, 2, 2, 6, 1, 10, 6, 14, 4, 18, 10, 22, 3, 26, 14, 30, 8, 34, 18, 38, 5, 42, 22, 46, 12, 50, 26, 54, 7, 58, 30, 62, 16, 66, 34, 70, 9, 74, 38, 78, 20, 82, 42, 86, 11, 90, 46, 94, 24, 98, 50, 102, 13, 106, 54, 110, 28, 114, 58

Offset: 0

Paul Curtz, May 22 2013

Repeated terms of A016825 are in the positions 1,2,3,6,5,10,... (A043547).
From Wolfdieter Lang, Dec 04 2013: (Start)
This sequence a(n), n>=1, appears in the formula 2*sin(2*Pi/n) = R(p(n), x) modulo C(a(n), x), with x = rho(a(n)) = 2*cos(Pi/a(n)), the R-polynomials given in A127672 and the minimal C-polynomials of rho given in A187360. This follows from the identity 2*sin(2*Pi/n) = 2*cos(Pi*p(n)/a(n)) with gcd(p(n), a(n)) = 1. For p(n) see a comment on A106609,
Because R is an integer polynomial it shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(a(n))) of degree delta(a(n)) (the degree of C(a(n), x)), with delta(k) = A055034(k). This degree is given in A093819. For the coefficients of 2*sin(2*Pi/n) in the power basis of Q(rho(a(n))) see A231189 . (End)

			For the first formula: a(0)=-1+1=0, a(1)=-3+5=2, a(2)=-1+3=2, a(3)=-1+7=6, a(4)=0+1=1.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).

Cf. A016825, A017089, A017137, A022998, A106609, A145979, A226008.

a[0]=0; a[n_] := Sqrt[Denominator[1/4 - 4/n^2]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, May 30 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,2,2,6,1,10,6,14,4,18,10,22,3,26,14,30},60] (* Harvey P. Dale, Nov 21 2019 *)

a(n) = A106609(n-4) + A106609(n+4) with A106609(-4)=-1, A106609(-3)=-3, A106609(-2)=-1, A106609(-1)=-1.
a(n) = 2*a(n-8) -a(n-16).
a(2n+1) = A016825(n), a(2n) = A145979(n-2) for n>1, a(0)=0, a(2)=2.
a(4n)   = A022998(n).
a(4n+1) = A017089(n).
a(4n+2) = A016825(n).
a(4n+3) = A017137(n).
G.f.: x*(2 +2*x +6*x^2 +x^3 +10*x^4 +6*x^5 +14*x^6 +4*x^7 +14*x^8 +6*x^9 +10*x^10 +x^11 +6*x^12 +2*x^13 +2*x^14)/((1-x)^2*(1+x)^2*(1+x^2)^2*(1+x^4)^2). [Bruno Berselli, May 23 2013]
From Wolfdieter Lang, Dec 04 2013: (Start)
a(n) = 2*n if n is odd; if n is even then a(n) is n if n/2 == 1, 3, 5, 7 (mod 8), it is n/2 if n/2 == 0, 4 (mod 8) and it is n/4 if n/2 == 2, 6 (mod 8). This leads to the given G.f..
With c(n) = A178182(n), n>=1, a(n) = c(n)/2 if c(n) is even and c(n) if c(n) is odd. This leads to the preceding formula. (End)

Edited by Bruno Berselli, May 24 2013

A228786 Table of coefficients of the minimal polynomials of 2*sin(Pi/n), n >= 1.

Original entry on oeis.org

0, 1, -2, 1, -3, 0, 1, -2, 0, 1, 5, 0, -5, 0, 1, -1, 1, -7, 0, 14, 0, -7, 0, 1, 2, 0, -4, 0, 1, -3, 0, 9, 0, -6, 0, 1, -1, 1, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 1, 0, -4, 0, 1, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, 1, -2, -1, 1, 1, 0, -8, 0, 14, 0, -7, 0, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1, 1, -3, 0, 1

Offset: 1

Wolfdieter Lang, Oct 07 2013

s(n) := 2*sin(Pi/n) is, for n >= 2, the length ratio side/R of the regular n-gon inscribed in a circle of radius R. This algebraic number s(n), n >= 1, has the degree gamma(n) := A055035(n), and the row length of this table is gamma(n) + 1.
s(n) has been given in the power basis of the relevant algebraic number field in A228783 for even n (bisected into n == 0 (mod 4) and n == 2 (mod 4)), and in A228785 for odd n.
For the computation of the minimal polynomials ps(n,x), using the coefficients of s(n) in the relevant number field, and the conjugates of the corresponding algebraic numbers rho (giving the length ratios (smallest diagonal)/side in the relevant regular polygons see a comment on A228781. Note that the product of the gamma(n) linear factors (x - conjugates) has to be computed modulo the minimal polynomial of the relevant rho(k) = 2*cos(Pi/k) (called C(k,x=rho(k)) in A187360).
Thanks go to Seppo Mustonen, who asked a question about the square of the sum of all lengths in the regular n-gon, which led to this computation of s(n) and its minimal polynomial.
It would be interesting to find out which length ratios in the regular n-gon give the other positive zeros of the minimal polynomial ps(n,x). See some examples below.
The zeros of the row polynomials ps(n,x) are 2*cos(2*Pi*k/c(2*n))) for gcd(k, c(2*n)) = 1, where c(n) = A178182(n), and k from {0, ..., floor(c(2*n)/2)}, for n >= 1. The number of these solutions is gamma(n) = A055035(n). See the formula section. This results from the zeros of the minimal polynomials of sin(2*Pi/n), with coefficients given in A181872/A181873. - Wolfdieter Lang, Oct 30 2019

			The table a(n, m) starts:
n\m   0  1    2 3   4 5     6 7   8 9   10 12  13 14  15 16 17
1:    0  1
2:   -2  1
3:   -3  0    1
4:   -2  0    1
5:    5  0   -5 0   1
6:   -1  1
7:   -7  0   14 0  -7 0     1
8:    2  0   -4 0   1
9:   -3  0    9 0  -6 0     1
10:  -1  1    1
11: -11  0   55 0 -77 0    44 0 -11 0    1
12:   1  0   -4 0   1
13:  13  0  -91 0 182 0  -156 0  65 0  -13 0   1
14:   1 -2   -1 1
15:   1  0   -8 0  14 0    -7 0   1
16:   2  0  -16 0  20 0    -8 0   1
17:  17  0 -204 0 714 0 -1122 0 935 0 -442  0 119  0 -17  0  1
18:   1 -3    0 1
...
n = 19: [-19, 0, 285, 0, -1254, 0, 2508, 0, -2717, 0, 1729, 0, -665, 0, 152, 0, -19, 0, 1],
n = 20: [1, 0, -12, 0, 19, 0, -8, 0, 1]
n = 5: ps(5,x) = 5 -5*x^2 +1*x^4, with the zeros s(5) = sqrt(3 - tau), sqrt(2 + tau) = tau*s(5) and their negative values, where tau =rho(5) is the golden section. tau*s(5) is the length ratio diagonal/radius in the pentagon.
n = 7: ps(7,x) = -7 + 14*x^2 -7*x^4 + 1*x^6, with the positive zeros s(7) (side/R) about 0.868, s(7)*rho(7) (smallest diagonal/R) about 1.564, and s(7)*(rho(7)^2-1) (longer diagonal/R) about 1.950 in the heptagon inscribed in a circle with radius R.
n = 8: ps(8,x) = 2 -4*x^2 + x^4, with the positive zeros s(8) = sqrt(2-sqrt(2)) and rho(8) = sqrt(2+sqrt(2)) (smallest diagonal/side).
n = 10: ps(10,x) = -1 + x + x^2 with the positive zero s(10) = tau - 1 (the negative solution is -tau).

Cf. A187360, A055035, A228781, A228783, A228785.

a(n, m) = [x^m](minimal polynomial ps(n, x) of 2*sin(Pi/n) over the rationals), n >= 1, m = 0, ..., gamma(n), with gamma(n) = A055035(n).

ps(n,x) = Product_{k=0..floor(c(2*n)/n) and gcd(k, c(2*n)) = 1} (x - 2*cos(2*Pi*k/c(2*n)), with c(2*n) = A178182(2*n), for n >= 1. There are gamma(n) = A055035(n) zeros. - Wolfdieter Lang, Oct 30 2019

A181871 Coefficient array for integer polynomial version of minimal polynomials of sin(2*Pi/n). Rising powers of x.

Original entry on oeis.org

0, 2, 0, 2, -3, 0, 4, -2, 2, 5, 0, -20, 0, 16, -3, 0, 4, -7, 0, 56, 0, -112, 0, 64, -2, 0, 4, -3, 0, 36, 0, -96, 0, 64, 5, 0, -20, 0, 16, -11, 0, 220, 0, -1232, 0, 2816, 0, -2816, 0, 1024, -1, 2, 13, 0, -364, 0, 2912, 0, -9984, 0, 16640, 0, -13312, 0, 4096, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 224, 0, -448, 0, 256, 2, 0, -16, 0, 16, 17, 0, -816, 0, 11424, 0, -71808, 0, 239360, 0, -452608, 0, 487424, 0, -278528, 0, 65536, -3, 0, 36, 0, -96, 0, 64

Offset: 1

Wolfdieter Lang, Jan 14 2011

The sequence of row lengths of this array is A093819(n)+1: [2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, ...].
pi(n,x) := Sum_{m=0..d(n)} a(n,m)*x^m, n >= 1, is related to the (monic) minimal polynomial of sin(2*Pi/n), called Pi(n,x), by pi(n,x) = (2^d(n))*Pi(n,x), with the degree sequence d(n)=A093819(n), and Pi(n,x) is given in A181872/A181873.
  Pi(n,x)=Psi(c(n),x) with the minimal polynomials Psi(n,x) of cos(2*Pi/n), and c(n):=A178182(n).
The minimal polynomials of sin(2*Pi/n) are, e.g., treated in the Lehmer and Niven references. (Note the mistake in the Lehmer references explained in the W. Lang link.) The fundamental polynomials Psi(n,x) are also studied in the Watkins-Zeitlin reference, where a recurrence is given.
See A231188 for the (monic and integer) minimal polynomials of 2*sin(2*Pi/n). = Wolfdieter Lang, Nov 30 2013

			[0, 2], [0, 2], [-3, 0, 4], [-2, 2], [5, 0, -20, 0, 16], [-3, 0, 4], [-7, 0, 56, 0, -112, 0, 64], [-2, 0, 4], [-3, 0, 36, 0, -96, 0, 64], [5, 0, -20, 0, 16], ...
pi(2,x) = (2^1)*Pi(2,x) = 2*Psi(c(2),x) = 2*Psi(4,x) = 2*x.

I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons..

Wolfdieter Lang, A181872/A181873. Minimal polynomials for sin(2Pi/n), and pi(n,x) polynomials..
D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40 (3) (1933) 165-6.
W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.

Cf. A181877 (cos(2*Pi/n) case), A231188 (2*sin(2*Pi/n) case).

ro[n_] := (cc = CoefficientList[ p = MinimalPolynomial[ Sin[2*(Pi/n)], x], x]; 2^Exponent[p, x]*(cc/Last[cc])); Flatten[ Table[ ro[n], {n, 1, 18}]] (* Jean-François Alcover, Sep 28 2011 *)

a(n,m) = [x^m]pi(n,x), n >= 1, m=0..A093819(n), and pi(n,x) defined above in the comments.

A327921 Irregular triangle T read by rows: row n gives the values determining the zeros of the minimal polynomial ps(n, x) of 2*sin(Pi/n) (coefficients in A228786), for n >= 1.

Original entry on oeis.org

1, 0, 1, 5, 1, 3, 1, 3, 7, 9, 1, 1, 3, 5, 9, 11, 13, 1, 3, 5, 7, 1, 5, 7, 11, 13, 17, 1, 2, 1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 1, 5, 7, 11, 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 1, 3, 5, 1, 7, 11, 13, 17, 19, 23, 29, 1, 3, 5, 7, 9, 11, 13, 15

Offset: 1

Wolfdieter Lang, Nov 02 2019

The minimal polynomials of the algebraic number s(n) = 2*sin(Pi/n) of degree gamma(n) = A055035(n) = A093819(2*n) has the zeros 2*cos(2*Pi*T(n,m)/c(2*n)), with c(2*n) = A178182(2*n), for m = 1, 2, ..., gamma(n) and n >= 1.
The number s(n) is the length ratio side(n)/R of the regular n-gon inscribed in a circle of radius R.
The motivation to look at these zeros came from the book of Carl Schick, and the paper by Brändli and Beyne. There, only length ratios diagonals/R in 2*(2*m + 1)-gons, for m >= 1, are considered.
If one is interested in length ratios diagonals/side then the minimal polynomials of rho(n) := 2*cos(Pi/n) (smallest diagonal/side) are important. These are given in A187360, called there C(n, x).

			The irregular triangle T(n,m) begins:
   n\m   1 2  3  4  5  6  7  8  9 10 11 12 ...      A178182(2*n)  A055035(n)
   -------------------------------------------------------------------------
   1:    1                                                4            1
   2:    0                                                1            1
   3:    1 5                                             12            2
   4:    1 3                                              8            2
   5:    1 3  7  9                                       20            4
   6:    1                                                6            1
   7:    1 3  5  9 11 13                                 28            6
   8:    1 3  5  7                                       16            4
   9:    1 5  7 11 13 17                                 36            6
  10:    1 2                                              5            2
  11:    1 3  5  7  9 13 15 17 19 21                     44           10
  12:    1 5  7 11                                       24            4
  13:    1 3  5  7  9 11 15 17 19 21 23 25               52           12
  14:    1 3  5                                          14            3
  15:    1 7 11 13 17 19 23 29                           60            8
  16:    1 3  5  7  9 11 13 15                           32            8
  ...
--------------------------------------------------------------------------
Some zeros are:
n = 1:  2*cos(2*Pi*1/4) = 0 = s(1),
n = 2:  2*cos(2*Pi*1/4) = 2 = s(2) (diameter/R),
n = 3:  2*cos(2*Pi*1/12) = -2*cos(2*Pi*5/12) = sqrt(3) = s(3),
n = 5:  2*cos(2*Pi*1/20) = -2*cos(2*Pi*9/20) = sqrt(2 + tau),
        2*cos(2*Pi*3/20) = -2*cos(2*Pi*7/20) = sqrt(tau - 3) = s(5),
with the golden ratio tau = A001622,
n = 10: 2*cos(2*Pi*1/5) = tau - 1 = s(10),  -2*cos(2*Pi*2/5) = -tau.
--------------------------------------------------------------------------

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, ISBN 3-9522917-0-6, Bobos Druck, Zürich, 2003.

Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757v2 [math.NT], 2015 and 2016.

Cf. A001622, A049310, A055035, A093819, A127672, A178182, A187360, A228786.

Row n gives the first gamma(n) = A055035(n) members of RRS(c(2*n)), for n >= 1, where RRS(k) is the smallest nonnegative restricted residue system modulo k.
The numbers with odd c(2*n) are n = 2 + 8*k, k >= 0.
The zeros x0^{(n)}_m := 2*cos(2*Pi*T(n,m)/c(2*n)) can be written as polynomials of rho(n) := 2*cos(Pi/n) for even n, and as polynomials of rho(2*n) for odd n as follows. x0^{(n)}_m = R(t*T(n,m), rho(b*n)), with b = 1 or 2 for n even or odd, respectively, and t = 1 for n == 1 (mod 2) and 0 (mod 4), t = 2 and 4 for n == 6 and 2 (mod 8), respectively. Here the monic Chebyshev T polynomials R(n, x) enter, with coefficients given in A127672. This results from 2*n/c(2*n) = 4, 2, 1, 1/2 for n == 2, 6 (mod 8), 0 (mod 4), 1 (mod 2), respectively. Note that rho(n)^2 = 4 - s(n)^2.
In terms of s(n) = 2*sin(Pi/n) the zeros x0^{(n)}_m are written with Chebyshev S (A049310) and R polynomials (A127672) as follows.
  x0^{(n)}_m = sqrt(4 - s(b*n)^2) * {S((T(n,m)-1)/2, -R(2, s(bn))) - S((T(n,m)-3)/2, -R(2, s(b*n)))}, for n == 1 (mod 2) with b(n) = 2, and for n == 0 (mod 4) with b = 1,
  x0^{(n)}_m = (2 - s(n)^2) * {S((T(n,m)-1)/2, R(4, s(n))) - S((T(n,m)-3)/2, R(4, s(n)))}, for n == 6 (mod 8), and
  x0^{(n)}_m = R(T(n,m), R(4, sqrt(4 - s(n)^2))), for n == 2 (mod 8).

cp's OEIS Frontend

A093819 Algebraic degree of sin(2*Pi/n).

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A181872 Numerators of coefficient array for minimal polynomials of sin(2*Pi/n). Rising powers of x.

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A225975 Square root of A226008(n).

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A228786 Table of coefficients of the minimal polynomials of 2*sin(Pi/n), n >= 1.

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A181871 Coefficient array for integer polynomial version of minimal polynomials of sin(2*Pi/n). Rising powers of x.

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A327921 Irregular triangle T read by rows: row n gives the values determining the zeros of the minimal polynomial ps(n, x) of 2*sin(Pi/n) (coefficients in A228786), for n >= 1.

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