A193681 - OEIS

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

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OFFSET

1,4

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 phi(2*n)/2) / determinant (A193681) .)

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).

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