A114654 Discriminant of the polynomial x^n + x + 1.

1, -3, -31, 229, 3381, -43531, -870199, 15953673, 404197705, -9612579511, -295311670611, 8630788777645, 311791207040509, -10809131718965763, -449005897206417391, 18008850183328692241, 845687005960046315793, -38519167813410200811247

Offset: 1

Author

T. D. Noe, Dec 21 2005

Keywords

• sign

Comments

Except for the sign, the sequence alternates between the sum and difference of consecutive terms of A000312. x^2+x+1 divides x^n+x+1 for n=2 (mod 3).

References

• Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Crossrefs

Cf. A000312 (n^n), A007781 (n^n - (n-1)^(n-1)), A056788 (n^n + (n-1)^(n-1)), A086797 (discriminant of the polynomial x^n-x-1).

Programs

Mathematica

Table[Discriminant[x^n + x + 1, x], {n, 0, 100}] (* Artur Jasinski, Oct 12 2007 *)

PARI

a(n) = poldisc(x^n+x+1); \\ Michel Marcus, Aug 28 2020

Formula

for n>1, a(n) = (n^n + (-1)^(n-1) * (n-1)^(n-1)) * (-1)^floor(n/2).

a(n) = (Cos[Pi n/2]+Sin[Pi n/2])(n^n)+(Cos[Pi(n+1)/2]+Sin[Pi(n+1)/2])(n+1)^(n+1). - Artur Jasinski, Oct 12 2007