A093819 -id:A093819 - cp's OEIS frontend

A055035 Degree of minimal polynomial of sin(Pi/n) over the rationals.

1, 1, 2, 2, 4, 1, 6, 4, 6, 2, 10, 4, 12, 3, 8, 8, 16, 3, 18, 8, 12, 5, 22, 8, 20, 6, 18, 12, 28, 4, 30, 16, 20, 8, 24, 12, 36, 9, 24, 16, 40, 6, 42, 20, 24, 11, 46, 16, 42, 10, 32, 24, 52, 9, 40, 24, 36, 14, 58, 16, 60, 15, 36, 32, 48, 10, 66, 32, 44, 12, 70, 24, 72

Offset: 1

Shawn Cokus (Cokus(AT)math.washington.edu)

Also degree of minimal polynomial of function F(n)=(gamma(1/n)*gamma((n-1)/n))/Pi over the rationals. Roots of minimal polynomials of F(n) belonging to algebraic extension of sin(n/Pi) and vice versa (e.g. gamma(1/11)*gamma(10/11)/Pi = 20*sin(Pi/11) - 112*sin(Pi/11)^3 + 256*sin(Pi/11)^5 - 256*sin(Pi/11)^7 + (1024*sin(Pi/11)^9)/11). - Artur Jasinski, Oct 17 2011

The algebraic numbers sin(Pi/(2*l)) are given in A228783 in the power basis of the number field Q(2*cos(Pi/(2*l))) if n is even and of Q(2*cos(Pi/l)) if l is odd. In A228785, sin(Pi/(2*l+1)) is given in the power basis of Q(2*cos(Pi/(2*(2*l+1)))) (only odd powers appear). The minimal polynomials for 2*sin(Pi/n), n>=1, are given in A228786. - Wolfdieter Lang, Oct 10 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..175
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. A000010, A228786 (row length), A093819.

a[n_] := If[n==2, 1, EulerPhi[n]/{1, 1, 2, 1}[[Mod[n, 4]+1]]]; Table[a[n], {n, 80}]
a[n_] := Exponent[ MinimalPolynomial[Sin[Pi/n]][x], x]; Array[a, 75] (* Robert G. Wilson v, Jul 28 2014 *)

a(1)=1, a(2)=1, a(n)=phi(n)/(1, 1, 2, 1 for n=0, 1, 2, 3 mod 4) for n>2, where phi is Euler's totient, A000010

a(n) = A093819(2*n), n >= 1.- Wolfdieter Lang, Oct 29 2019

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

A231190 Numerator of abs(n-8)/(2*n), n >= 1.

Original entry on oeis.org

7, 3, 5, 1, 3, 1, 1, 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 4, 65, 33, 67, 17

Offset: 1

Wolfdieter Lang, Dec 12 2013

Because 2*sin(Pi*4/n) = 2*cos(Pi*abs(n-8)/(2*n)) = 2*cos(Pi*a(n)/b(n)) with gcd(a(n),b(n)) = 1, one has
  2*sin(Pi*4/n) = R(a(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/b(n)) =: rho(b(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively.
  b(n) = A232625(n). This shows that 2*sin(Pi*4/n) is an integer in the algebraic number field Q(rho(b(n))) of degree delta(b(n)), with delta(k) = A055034(k). This degree delta(b(n)) is given in A231193(n), and if gcd(n,2) = 1 it coincides with the one for sin(2*Pi/n) given by A093819(n). See Theorem 3.9 of the I. Niven reference, pp. 37-38, which uses gcd(k, n) = 1. See also the Jan 09 2011 comment on A093819.
a(n) and b(n) = A232625(n) are the k=2 members of a family of pair of sequences p(k,n) and q(k,n), n >= 1, k >= 1, relevant to determine the algebraic degree of 2*sin(Pi*2*k/n) from the trigonometric identity (used in the D. H. Lehmer and I. Niven references) 2*sin(Pi*2*k/n) = 2*cos(Pi*abs(n-4*k)/(2*n)) = 2*cos(Pi*p(k,n)/q(k,n)). This is R(p(k,n), x) (mod C(q(k,n), x)), with x = 2*cos(Pi/q(k,n)) =: rho(q(k,n)). The polynomials R and C have been used above. C(q(k,n), x) is the minimal polynomial of rho(q(k,n)) with degree delta(q(k,n)), which is then the degree, call it deg(k,n), of the integer 2*sin(Pi*2*k/n) in the number field Q(rho(q(k,n))). From Theorem 3.9 of the I. Niven reference deg(k,n) is, for given k, for those n with gcd(k, n) = 1 determined by A093819(n). In general deg(k,n) = A093819(n/gcd(k,n)). For the k=1 instance p(1,n) and q(1,n) see comments on A106609 and A225975.

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

Robert Israel, Table of n, a(n) for n = 1..10000
D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.

Cf. A127672 (R), A187360 (C), A232625 (b), A055034 (delta), A093819 (degree if k=1), A232626(degree if k=2), A106609 (k=1, p), A225975 (k=1, q), A106617.

f:= n -> numer(abs(n-8)/(2*n)):
map(f, [$1..100]); # Robert Israel, Dec 06 2018

a[n_] := Numerator[Abs[n-8]/(2n)]; Array[a, 50] (* Amiram Eldar, Dec 06 2018 *)

a(n) = numerator(abs(n-8)/(2*n)), n >= 1.
a(n) = abs(n-8)/gcd(n-8, 16).
a(n) = abs(n-8) if n is odd; if n is even then a(n) = abs(n-8)/2 if n/2 == 1, 3, 5, 7 (mod 8), a(n) = abs(n-8)/4  if n/2 == 2, 6 (mod 8), a(n) = abs(n-8)/8 if n/2 == 0 (mod 8) and a(n) = abs(n-8)/16 if n == 4 (mod 8).
O.g.f.: 1+ x*(7 + 3*x + 5*x^2 + 1*x^3 + 3*x^4 + 1*x^5 + 1*x^6) + N(x)/(1-x^16)^2 , with N(x) = x^9*((1+x^30) + x*(1+x^28)  + 3*x^2*(1+x^26) + x^3*(1+x^24) + 5*x^4*(1+x^22) + 3*x^5*(1+x^20) + 7*x^6*(1+x^18) + x^7*(1+x^16) + 9*x^8*(1+x^14) + 5*x^9*(1+x^12) + 11*x^10*(1+x^10) + 3*x^11*(1+x^8) + 13*x^12*(1+x^6) + 7*x^13*(1+x^4) + 15*x^14*(1+x^2)+x^15).
a(n+32)-2*a(n+16)+a(n) = 0 for n >= 8.
a(n+8) = A106617(n). - Peter Bala, Feb 28 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

A231188 Coefficient table for the minimal polynomials of 2sin(2Pi/n). Rising powers of x.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 29 2013

The length of row n is deg(n) + 1 = A093819(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, 2, 13, 7, 9, 5, 17,...
See A181871 for the coefficient table for the integer but non-monic minimal polynomials of sin(2*Pi/n), n>=1, called there pi(n, x). The present minimal polynomials of 2*sin(2*Pi/n) are integer and monic, and they are given by
  MP2sin2(n, x) = pi(n, x/2).

			The table a(n,m) starts:
---------------------------------------------------------------------------------
n\m   0   1    2  3    4  5     6  7    8  9    10  11   12  13   14 15 16 ...
1:    0   1
2:    0   1
3:   -3   0    1
4:   -2   1
5:    5   0   -5  0    1
6:   -3   0    1
7:   -7   0   14  0   -7  0     1
8:   -2   0    1
9:   -3   0    9  0   -6  0     1
10:   5   0   -5  0    1
11: -11   0   55  0  -77  0    44  0  -11  0     1
12:  -1   1
13:  13   0  -91  0  182  0  -156  0   65  0   -13   0    1
14:  -7   0   14  0   -7  0     1
15:   1   0   -8  0   14  0    -7  0    1
16:   2   0   -4  0    1
17:  17   0 -204  0  714  0 -1122  0  935  0  -442   0  119   0  -17  0  1
...

Cf. A093819, A181871, A181872/A181872, A232624.

a(n,m) = [x^m] MP2sin2(n, x), n>=1, m = 0, 1, ..., A093819(n), with the minimal polynomials of 2*sin(2*Pi/n), given above in a comment in terms of the ones for sin(2*Pi/n).

A232625 Denominators of abs(n-8)/(2*n), n >= 1.

Original entry on oeis.org

2, 2, 6, 2, 10, 6, 14, 1, 18, 10, 22, 6, 26, 14, 30, 4, 34, 18, 38, 10, 42, 22, 46, 3, 50, 26, 54, 14, 58, 30, 62, 8, 66, 34, 70, 18, 74, 38, 78, 5, 82, 42, 86, 22, 90, 46, 94, 12, 98, 50, 102, 26, 106, 54, 110, 7, 114, 58, 118, 30, 122, 62, 126, 16, 130, 66

Offset: 1

Wolfdieter Lang, Dec 12 2013

The numerators are given in A231190. See the comments there on 2*sin(Pi*4/n).
2*sin(Pi*4/n) = R(b(n), x) (mod C(b(n), x)), with x = 2*cos(Pi/a(n)) =: rho(a(n)). The integer Chebyshev R and C polynomials are found in A127672 and A187360, respectively.  b(n) = A231190(n).
delta(a(n)) = deg(2,n), with delta(k) = A055034(k), is the degree of the algebraic number 2*sin(Pi*4/n) given in A232626.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A127672 (R), A187360 (C), A231190 (b), A055034 (delta), A232626 (degree k=2), A106609 (k=1, p), A225975 (k=1, q), A093819 (degree k=1).

a[n_] := Denominator[(n-8)/(2*n)]; Array[a, 100] (* Amiram Eldar, Nov 09 2024 *)

a(n) = denominator((n-8)/(2*n)); \\ Amiram Eldar, Nov 09 2024

a(n) = denominator(abs(n-8)/(2*n)), n >= 1.
a(n) = 2*n/gcd(n-8, 16).
a(n) = 2*n if n is odd; if n is even then a(n) = n if n/2 == 1, 3, 5, 7 (mod 8), a(n) = n/2  if n/2 == 2, 6 (mod 8), a(n) == n/4 if n/2 == 0 (mod 8) and a(n) = n/8 if n == 4 (mod 8).
O.g.f.: x*(2*(1+x^30) + 2*x*(1+x^28) + 6*x^2*(1+x^26) + 2*x^3*(1+x^24) + 10*x^4*(1+x^22) + 6*x^5*(1+x^20) + 14*x^6*(1+x^18) + x^7*(1+x^16) + 18*x^8*(1+x^14) + 10*x^9*(1+x^12) + 22*x^10*(1+x^10) + 6*x^11*(1+x^8) + 26*x^12*(1+x^6) + 14*x^13*(1+x^4) + 30*x^14*(1+x^2) + 4*x^15)/(1-x^16)^2.
Sum_{k=1..n} a(k) ~ (171/256) * n^2. - Amiram Eldar, Nov 09 2024

A232626 Degree of the algebraic number 2sin(4Pi/n).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Dec 12 2013

See the comment on A231190 for the formula for 2*sin(Pi*4/n) = 2*cos(Pi*p(2,n)/q(2,n)) with gcd(p(2,n),q(2,n)) = 1, where p(2,n) = A231190(n) and q(2,n) = A232625(n). This shows that 2*sin(Pi*4/n) is an integer in the algebraic number field Q(rho(q(2,n))) of degree a(n) = delta(q(2,n)) with delta(k) = A055034(k).

This degree a(n) is given by I. Niven's Theorem 3.9, pp. 37-38, by Niven(n/gcd(2,n)) with Niven(n) = A093819(n) the degree of 2*sin(2*Pi/n). Note that Niven uses gcd(k, n) = 1 in the derivation, and Niven(4) = 1. See the bisection given in the formula section which is obtained from this.

			a(1) = A093819(1) = 1; a(4) = phi(2) = 1; a(6) = phi(3) = 2; a(8) = 1; a(9) = A093819(9) = 6.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..175 from Vincenzo Librandi)

Cf. A000010, A055034, A231190, A232625, A093819.

f[n_] := Exponent[ MinimalPolynomial[ 2Sin[ 4Pi/n]][x], x]; Array[f, 75] (* Robert G. Wilson v, Jul 28 2014 *)

a(n) = {my(k = denominator((n-8)/(2*n))); if(k == 1, 1, eulerphi(2*k)/2);} \\ Amiram Eldar, Nov 09 2024

a(n) = delta(A232625(n)), n >=1, with delta(1) = 1 and delta(k) = phi(2*k)/2 with Euler's totient function phi (A000010). delta(k) = A055034(k).
a(2*k+1) = A093819(2*k+1), k >= 0.
For k >= 1: a(2*k) = A093819(k), that is a(2*k) = 1 if k=4, phi(k) if k odd or k == 2 (mod 4), phi(k)/2 if k == 0 (mod 8), phi(k)/4 if k == 4 (mod 8) (but not k=4).

A181873 Denominators of coefficient array for minimal polynomials of sin(2Pi/n). Rising powers of x.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 1, 1, 16, 1, 4, 1, 1, 4, 1, 1, 64, 1, 8, 1, 4, 1, 1, 2, 1, 1, 64, 1, 16, 1, 2, 1, 1, 16, 1, 4, 1, 1, 1024, 1, 256, 1, 64, 1, 4, 1, 4, 1, 1, 2, 1, 4096, 1, 1024, 1, 128, 1, 16, 1, 16, 1, 4, 1, 1, 64, 1, 8, 1, 4, 1, 1, 256, 1, 8, 1, 8, 1, 4, 1, 1, 8, 1, 1, 1, 1, 65536, 1, 4096, 1, 2048, 1, 512, 1, 256, 1, 32, 1, 16, 1, 4, 1, 1, 64, 1, 16, 1, 2, 1, 1, 262144, 1, 65536, 1, 8192, 1, 1024, 1, 1024, 1, 256, 1, 64, 1, 2, 1, 4, 1, 1, 4, 2, 1, 4096, 1, 64, 1, 64, 1, 32, 1, 4, 1, 4, 1, 1, 1024, 1, 256, 1, 64, 1, 4, 1, 4, 1, 1

Offset: 1

Wolfdieter Lang, Jan 13 2011

The corresponding numerator array is given in A181872(n,m) where details, references, and a W. Lang link are given.

The sequence of row lengths of this array is d(n)+1 with d(n)=A093819(n): [2, 2, 3, 4, 5, 3, 7, 3, 7, 5, 11,... ].

			[1, 1], [1, 1], [4, 1, 1], [1, 1], [16, 1, 4, 1, 1], [4, 1, 1], [64, 1, 8, 1, 4, 1, 1], [2, 1, 1], [64, 1, 16, 1, 2, 1, 1], [16, 1, 4, 1, 1],...
The rational coefficients A181872(n,m)/a(n,m) start with:
[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],...

See A181872.

See A181872.

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

Cf. A181872.

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

a(n,m)=denominator([x^m]Pi(n,x)), n>=1, m=0,1,...,d(n), with the d(n)=A093819(n), and Pi(n,x) the minimal polynomials of sin(2*Pi/n) given in A181872.

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.

A232629 Coefficients of the algebraic number 2sin(4Pi/n) in the power basis of Q(2*cos(Pi/q(2,n))), with q(2,n) = A232625(n), n >= 1.

Original entry on oeis.org

0, 0, 0, -1, 0, 0, -3, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1, 0, 0, 0, -7, 0, 14, 0, -7, 0, 1, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, -5, 0, 1, 0, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1

Offset: 1

Wolfdieter Lang, Dec 17 2013

The length of row n is A232626(n).
In a regular n-gon, n>=2, inscribed in a circle of radius R (in some length units), 2*sin(4*Pi/n) = (S(n)/R)*(D(1,n)/S(n)) = D(1,n)/R, with the side length S(n) and the length of the first (smallest) diagonal D(1,n). For n=2 there is no such diagonal, and one can put D(1,2) = 0. Obviously, D(1,2*m) = S(m), m >= 2.
See a comment on A231190 regarding the pair of sequences  p(k,n) and q(k,n), n >= 1, k >= 1. Here k=2 with A231190 and A232625.
See also the k=1 analog A231189 of the present table.
The relevant identity is here 2*sin(Pi*4/n) = 2*cos(Pi*abs(n-8)/(2*n)) = 2*cos(Pi*p(2,n)/q(2,n)). This is R(p(2,n), x) (mod C(q(2,n), x)), with x = 2*cos(Pi/q(2,n)) =: rho(q(2,n)). with the coefficient tables for the polynomials R and C given in A127672 and A187360, respectively. This gives the power base coefficients of 2*sin(Pi*4/n) in the algebraic number field Q(rho(q(2,n))) of degree delta(q(2,n)), with delta(n) = A055034(n), shown in A232626.
If the degree p(2,n) of R(p(2,n), x) is smaller than the degree A232626(n) of C(q(2,n), x) then 2*sin(Pi*4/n) = R(p(2,n), x). Otherwise the (mod C(q(2,n), x)) congruence is needed. This happens for n = 1, 2, 3, 4, 21, 24, 27, 30,...
The power basis of Q(rho(q(2,n))) is <1, rho(q(2,n)), ..., rho(q(2,n))^(delta(q(2,n))-1)>. Therefore the length of row n of this table is delta(q(2,n)) = A232626(n).
The coefficient table for the minimal polynomial of 2*sin(Pi*4/n) is given in A232630.

			The table a(n,m) begins (the trailing zeros are needed to have the correct degree A232626(n) in Q(rho(q(2,n))))
-----------------------------------------------------------------
n\m 0   1  2   3  4   5  6   7  8   9 10 11 12 13 14 15 16 17 ...
1:  0
2:  0
3:  0  -1
4:  0
5:  0  -3  0   1
6:  0   1
7:  0   1  0   0  0   0
8:  2
9:  0   1  0   0  0   0
10: 0   1  0   0
11: 0  -3  0   1  0   0  0   0  0   0
12: 0   1
13: 0   5  0  -5  0   1  0   0  0   0  0  0
14: 0  -3  0   1  0   0
15: 0  -7  0  14  0  -7  0   1
16: 0   1
17: 0   9  0 -30  0  27  0  -9  0   1  0  0  0  0  0  0
18: 0   5  0  -5  0   1
19: 0 -11  0  55  0 -77  0  44  0 -11  0  1  0  0  0  0  0  0
20: 0  -3  0   1
...
n=1: 2*sin(Pi*4/1) = 0. R(p(2,1), x) = R(7, x) = -7*x + 14*x^3 -7*x^5 + x^7. C(q(2,1), x) = C(2, x) = x, hence R(7, x) (mod C(2, x)) == 0, and
  with A232626(1) = 1, a(1,0) = 0.n=7: p(2,7) = A231190(7) = 1, q(2,7) = A232625(7) = 14. 2*sin(Pi*4/7) = R(1, x) = x = rho(14) := 2*cos(Pi/14). C(14, x) of degree 6 does not apply here. A232626(7) = 6, hence the row n=7 is  0  1  0  0  0  0.
n=9: p(2,9) = 1, q(2,9) = 18. 2*sin(Pi*4/9) = R(1, x) = x = rho(18) = 2*cos(Pi/18). C(18, x) with degree 6 is not needed here. A232626(9) = 6, hence row n=9 is also 0  1  0  0  0  0.
n=8: this row with entry 2 coincides with row n=4 of A231189.
n=17: row length A232626(17) = 16; p(2,17) = 9; C(34, x) has degree 16, therefore the R(9, x) coefficients produce here the first 10 entries for row n=17: 0  9  0 -30  0  27  0 -9  0  1, followed by 6 zeros, and 2*sin(Pi*4/17) = 9*rho(34) - 30*rho(34)^3 + 27*rho(34)^5 - 9*rho(34)^7 + 1*rho(34)^9, with rho(34) = 2*cos(Pi/34).

Cf. A231190 (p), A232625 (q), A127672 (R), A187360 (C), A232626 (degree), A231189 (k=1 case), A232630 (minimal polynomials).

a(n,m) = [x^m] (R(p(2,n), x) (mod C(q(2,n), x)), n >= 1, m = 0, 1, ..., A232626(n) - 1, where the C and R polynomials are found in A187360 and A127672, respectively. p(2,n) = A231190(n) and q(2,n) = A232625(n). Powers of x = rho(q(2,n)) := 2*cos(Pi/q(2,n)) appear in the table in increasing order.

a(2*l,m) = A231189(l,m), l >= 1, m = 0, 1, ..., (A093819(n)-1).

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A055035 Degree of minimal polynomial of sin(Pi/n) over the rationals.

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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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A231190 Numerator of abs(n-8)/(2*n), n >= 1.

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A225975 Square root of A226008(n).

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A231188 Coefficient table for the minimal polynomials of 2*sin(2*Pi/n). Rising powers of x.

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A232625 Denominators of abs(n-8)/(2*n), n >= 1.

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A232626 Degree of the algebraic number 2*sin(4*Pi/n).

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A181873 Denominators of coefficient array for minimal polynomials of sin(2Pi/n). Rising powers of x.

A231188 Coefficient table for the minimal polynomials of 2sin(2Pi/n). Rising powers of x.

A232626 Degree of the algebraic number 2sin(4Pi/n).

A232629 Coefficients of the algebraic number 2sin(4Pi/n) in the power basis of Q(2*cos(Pi/q(2,n))), with q(2,n) = A232625(n), n >= 1.