1,2

This polynomial is the characteristic polynomial of the Fibonacci and Lucas n-step sequences. These discriminants are prime for n=2, 4, 6, 26, 158 (A106274). It appears that the term a(2n+1) always has a factor of 2^(2n). With that factor removed, the discriminants are prime for odd n=3, 5, 7, 21, 99, 405. See A106275 for the combined list.

Eric Weisstein's World of Mathematics, Fibonacci n-Step

Eric Weisstein's World of Mathematics, Polynomial Discriminant

a(n) = (-1)^(n(n+1)/2) * ((n+1)^(n+1)-2(2n)^n)/(n-1)^2 - Max Alekseyev (maxal(AT)cs.ucsd.edu), May 05 2005

Discriminant[p_?PolynomialQ, x_] := With[{n=Exponent[p, x]}, Cancel[((-1)^(n(n-1)/2) Resultant[p, D[p, x], x])/Coefficient[p, x, n]^(2n-1)]]; Table[Discriminant[x^n-Sum[x^i, {i, 0, n-1}], x], {n, 20}]

(PARI) {a(n)=(-1)^(n*(n+1)/2)*((n+1)^(n+1)-2*(2*n)^n)/(n-1)^2} (Alekseyev)

Cf. A086797 (discriminant of the polynomial x^n-x-1), A000045, A000073, A000078, A001591, A001592 (Fibonacci n-step sequences), A000032, A001644, A073817, A074048, A074584 (Lucas n-step sequences), A086937, A106276, A106277, A106278 (number of distinct zeros of these polynomials for n=2, 3, 4, 5).

Sequence in context: A109984 A096355 A054766 this_sequence A052803 A048940 A058792

Adjacent sequences: A106270 A106271 A106272 this_sequence A106274 A106275 A106276

sign

T. D. Noe (noe(AT)sspectra.com), May 02 2005