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

1,5

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,... ].

References

See A181872.

Links

Table of n, a(n) for n=1..156.

See A181872.

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.

Example

[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],...

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 *)

Crossrefs

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

Cf. A181872.

Sequence in context: A038025 A079982 A265143 * A229293 A269443 A039927

Adjacent sequences: A181870 A181871 A181872 * A181874 A181875 A181876

Keyword

nonn,easy,frac,tabf

Author

Wolfdieter Lang, Jan 13 2011

Status

approved