login
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
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
Ed Pegg Jr, Table illustrating A193681 (Each box gives n, degree (A055034), and determinant (this sequence).)
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
KEYWORD
nonn,easy,changed
AUTHOR
Wolfdieter Lang, Sep 13 2011
STATUS
approved