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

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

Author

Wolfdieter Lang, Jan 13 2011

Keywords

• nonn
• easy
• frac
• tabf

Comments

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,... ].

Examples

			[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],...
		

References

• See A181872.

Links

• See A181872.

Crossrefs

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

Cf. A181872.

Programs

Mathematica

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 *)

Formula

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.