cp's OEIS Frontend

A106273 Discriminant of the polynomial x^n - x^(n-1) - ... - x - 1.

%I A106273 #75 Apr 12 2025 16:06:12
%S A106273 1,5,-44,-563,9584,205937,-5390272,-167398247,6042477824,249317139869,
%T A106273 -11597205023744,-601139006326619,34383289858207744,
%U A106273 2151954708695291177,-146323302326154543104,-10742330662077208945103,846940331265064719417344,71373256668946058057974997
%N A106273 Discriminant of the polynomial x^n - x^(n-1) - ... - x - 1.
%C A106273 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.
%C A106273 a(n) is the determinant of an r X r Hankel matrix whose entries are w(i+j) where w(n) = x1^n + x2^n + ... + xr^n where x1,x2,...xr are the roots of the titular characteristic polynomial. E.g., A000032 for n=2, A001644 for n=3, A073817 for n=4, A074048 for n=5, A074584 for n=6, A104621 for n=7, ... - _Kai Wang_, Jan 17 2021
%C A106273 Luca proves that a(n) is a term of the corresponding k-nacci sequence only for n=2 and 3. - _Michel Marcus_, Apr 12 2025
%H A106273 Vincenzo Librandi, <a href="/A106273/b106273.txt">Table of n, a(n) for n = 1..200</a>
%H A106273 Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
%H A106273 Michael Baake and Uwe Grimm, <a href="https://arxiv.org/abs/1907.11012">Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings</a>, arXiv:1907.11012 [math.MG], 2019.
%H A106273 Herbert Batte and Florian Luca, <a href="https://arxiv.org/abs/2504.02514">The Discriminant of the Characteristic Polynomial of the kth Fibonacci sequence is not a member of the kth Lucas sequence</a>, arXiv:2504.02514 [math.NT], 2025.
%H A106273 Florian Luca, <a href="https://www.fq.math.ca/Papers/62-3/luca02082024.pdf">On the discriminant of the k-generalized Fibonacci polynomial, II</a>, Fibonacci Quart. 62 (2024), no. 3, 193-200.
%H A106273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H A106273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolynomialDiscriminant.html">Polynomial Discriminant</a>
%F A106273 a(n) = (-1)^(n*(n+1)/2) * ((n+1)^(n+1)-2*(2*n)^n)/(n-1)^2. - _Max Alekseyev_, May 05 2005
%t A106273 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}]
%o A106273 (PARI) {a(n)=(-1)^(n*(n+1)/2)*((n+1)^(n+1)-2*(2*n)^n)/(n-1)^2}  \\ _Max Alekseyev_, May 05 2005
%o A106273 (PARI) a(n)=poldisc('x^n-sum(k=0,n-1,'x^k)); \\ _Joerg Arndt_, May 04 2013
%Y A106273 Cf. A086797 (discriminant of the polynomial x^n-x-1), A000045, A000073, A000078, A001591, A001592 (Fibonacci n-step sequences), A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755 (Lucas n-step sequences), A086937, A106276, A106277, A106278 (number of distinct zeros of these polynomials for n=2, 3, 4, 5).
%K A106273 sign
%O A106273 1,2
%A A106273 _T. D. Noe_, May 02 2005