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


%S 1,1,1,8,5,12,49,2048,81,2000,14641,2304,371293,1075648,1125,

%T 2147483648,410338673,1259712,16983563041,1024000000,453789,

%U 2414538435584,41426511213649,1358954496,762939453125,7340688973975552,31381059609,4739148267126784,10260628712958602189,324000000

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

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

%C 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 Robert Israel, <a href="/A193681/b193681.txt">Table of n, a(n) for n = 1..500</a>

%H Ed Pegg Jr, <a href="/A193681/a193681.jpg">Table illustrating A193681</a> {Each box gives n , degree (A055034 phi(2*n)/2) / determinant (A193681) .)

%F 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 g:= proc(n) local P,z,j;

%p P:= factor(evala(Norm(z-convert(2*cos(Pi/n),RootOf))));

%p if type(P,`^`) then P:= op(1,P) fi;

%p discrim(P,z)

%p end proc:

%p map(g, [$1..100]); # _Robert Israel_, Aug 04 2015

%t Table[NumberFieldDiscriminant[Cos[Pi/m]], {m, 1, z}] (* _Clark Kimberling_, Aug 03 2015 *)

%Y Cf. A055034, A127670, A187360, A193678.

%K nonn,easy

%O 1,4

%A _Wolfdieter Lang_, Sep 13 2011

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Last modified February 29 02:35 EST 2020. Contains 332353 sequences. (Running on oeis4.)