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

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

Author

Wolfdieter Lang, Jan 13 2011

Keywords

• sign
• easy
• frac
• tabf

Comments

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

Examples

			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.
		

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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