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

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

Author

Wolfdieter Lang, Jan 14 2011

Keywords

• sign
• easy
• tabf

Comments

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

Examples

			[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.
		

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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