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

%I A193681 #24 Oct 22 2024 03:25:11
%S A193681 1,1,1,8,5,12,49,2048,81,2000,14641,2304,371293,1075648,1125,
%T A193681 2147483648,410338673,1259712,16983563041,1024000000,453789,
%U A193681 2414538435584,41426511213649,1358954496,762939453125,7340688973975552,31381059609,4739148267126784,10260628712958602189,324000000
%N A193681 Discriminant of minimal polynomial of 2*cos(Pi/n) (see A187360).
%C A193681 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.
%C A193681   The zeros of the polynomials C(n,x) with coefficient triangle A187360 are given there in a comment.
%C A193681 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.
%H A193681 Robert Israel, <a href="/A193681/b193681.txt">Table of n, a(n) for n = 1..500</a>
%H A193681 Ed Pegg Jr, <a href="/A193681/a193681.jpg">Table illustrating A193681</a> (Each box gives n, degree (A055034), and determinant (this sequence).)
%F A193681 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).
%p A193681 g:= proc(n) local P,z,j;
%p A193681    P:= factor(evala(Norm(z-convert(2*cos(Pi/n),RootOf))));
%p A193681    if type(P,`^`) then P:= op(1,P) fi;
%p A193681    discrim(P,z)
%p A193681 end proc:
%p A193681 map(g, [$1..100]); # _Robert Israel_, Aug 04 2015
%t A193681 Table[NumberFieldDiscriminant[Cos[Pi/m]], {m, 1, z}]  (* _Clark Kimberling_, Aug 03 2015 *)
%Y A193681 Cf. A055034, A127670, A187360, A193678.
%K A193681 nonn,easy
%O A193681 1,4
%A A193681 _Wolfdieter Lang_, Sep 13 2011