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A193681 Discriminant of minimal polynomial of 2*cos(Pi/n) (see A187360). 2
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 (list; graph; refs; listen; history; text; internal format)
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

MATHEMATICA

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

CROSSREFS

Cf. A055034, A127670, A187360, A193678.

Sequence in context: A316689 A087462 A168204 * A253806 A199806 A070484

Adjacent sequences:  A193678 A193679 A193680 * A193682 A193683 A193684

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 13 2011

STATUS

approved

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Last modified January 24 16:41 EST 2020. Contains 331208 sequences. (Running on oeis4.)