A127670 -id:A127670 - cp's OEIS frontend

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

Offset: 1

Wolfdieter Lang, Sep 13 2011

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.

Robert Israel, Table of n, a(n) for n = 1..500
Ed Pegg Jr, Table illustrating A193681 (Each box gives n, degree (A055034), and determinant (this sequence).)

Cf. A055034, A127670, A187360, A193678.

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

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

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).

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

Original entry on oeis.org

1, 1, -4, -32, 400, 6912, -153664, -4194304, 136048896, 5120000000, -219503494144, -10567230160896, 564668382613504, 33174037869887488, -2125764000000000000, -147573952589676412928, 11034809241396899282944, 884295678882933431599104, -75613185918270483380568064

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Fibonacci polynomials.

Fibonacci polynomials are defined as F(0)=0, F(1)=1 and F(n)=x*F(n-1)+F(n-2) for n>1. Coefficients are given in triangle A168561 with offset 1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A168561, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A127670.

[(-1)^((n-2)*(n-1) div 2)*2^(n-1)*n^(n-3): n in [1..20]]; // Vincenzo Librandi, Aug 27 2018

Array[(-1)^((#-2)*(#-1)/2)*2^(#-1)*#^(#-3)&,20]

concat([1], [poldisc(p) | p<-Vec(x/(1-x^2-y*x) - x + O(x^20))]) \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

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

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A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A317403 a(n)=(-1)^((n-2)*(n-1)/2)*2^(n-1)*n^(n-3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).