A193681 Discriminant of minimal polynomial of 2*cos(Pi/n) (see A187360).

1, 1, 1, 8, 5, 12, 49, 2048, 81, 2000, 14641, 2304, 371293, 1075648, 1125, 2147483648, 410338673, 1259712, 16983563041, 1024000000, 453789, 2414538435584, 41426511213649, 1358954496, 762939453125, 7340688973975552, 31381059609, 4739148267126784, 10260628712958602189, 324000000

Offset: 1

Author

Wolfdieter Lang, Sep 13 2011

Keywords

• nonn
• easy

Comments

For the discriminant of a polynomial in terms of the square of a determinant of a Vandermonde matrix build from the zeros of the polynomial see, e.g., A127670.

The zeros of the polynomials C(n,x) with coefficient triangle A187360 are given there in a comment.

The discriminant of the monic C(n,x) polynomial can also be computed from its zeros x_i and the derivative of C, via (-1)^binomial(delta(n),2)*product(C'(n,x)|_{x=x_i},i=1..delta(n)), with the degree delta(n)=A055034(n). For a reference see, e.g., Rivlin, p. 218, quoted in A127670.

Links

• Robert Israel, Table of n, a(n) for n = 1..500
• Ed Pegg Jr, Table illustrating A193681 (Each box gives n, degree (A055034), and determinant (this sequence).)

Crossrefs

Cf. A055034, A127670, A187360, A193678.

Programs

Maple

g:= proc(n) local P,z,j;
   P:= factor(evala(Norm(z-convert(2*cos(Pi/n),RootOf))));
   if type(P,`^`) then P:= op(1,P) fi;
   discrim(P,z)
end proc:
map(g, [$1..100]); # Robert Israel, Aug 04 2015

Mathematica

Table[NumberFieldDiscriminant[Cos[Pi/m]], {m, 1, z}]  (* Clark Kimberling, Aug 03 2015 *)

Formula

a(n) = discriminant(C(n,x)), n>=1, with C(n,x):=sum(A187360(n,m)*x^m,m=0..A055034(n)), the minimal polynomial of 2*cos(Pi/n).