A106277 -id:A106277 - cp's OEIS frontend

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

1, 5, -44, -563, 9584, 205937, -5390272, -167398247, 6042477824, 249317139869, -11597205023744, -601139006326619, 34383289858207744, 2151954708695291177, -146323302326154543104, -10742330662077208945103, 846940331265064719417344, 71373256668946058057974997

Offset: 1

T. D. Noe, May 02 2005

sign

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.
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
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

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
Michael Baake and Uwe Grimm, Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings, arXiv:1907.11012 [math.MG], 2019.
Herbert Batte and Florian Luca, The Discriminant of the Characteristic Polynomial of the kth Fibonacci sequence is not a member of the kth Lucas sequence, arXiv:2504.02514 [math.NT], 2025.
Florian Luca, On the discriminant of the k-generalized Fibonacci polynomial, II, Fibonacci Quart. 62 (2024), no. 3, 193-200.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Polynomial Discriminant

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).

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}]

{a(n)=(-1)^(n*(n+1)/2)*((n+1)^(n+1)-2*(2*n)^n)/(n-1)^2}  \\ Max Alekseyev, May 05 2005

a(n)=poldisc('x^n-sum(k=0,n-1,'x^k)); \\ Joerg Arndt, May 04 2013

a(n) = (-1)^(n*(n+1)/2) * ((n+1)^(n+1)-2*(2*n)^n)/(n-1)^2. - Max Alekseyev, May 05 2005

A106280 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 4 distinct zeros.

Original entry on oeis.org

137, 179, 653, 859, 991, 1279, 1601, 1609, 2089, 2437, 2591, 2693, 2789, 2897, 3701, 3823, 3847, 4451, 4691, 4751, 4919, 5431, 5479, 5807, 5903, 5953, 6203, 6421, 6781, 6917, 7253, 7867, 8317, 9187, 9277, 9533, 9629, 9767, 9907, 9967, 10009, 10079

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817. The periods of the sequences A000078(k) mod p and A073817(k) mod p have length less than p.

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

Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of 4-step sequence mod prime(n)).

t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 1600}];Prime[Flatten[Position[t, 4]]]

A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.

Original entry on oeis.org

2, 5, 11, 13, 31, 43, 53, 79, 83, 89, 97, 103, 109, 131, 139, 151, 197, 199, 229, 233, 239, 251, 257, 271, 283, 313, 317, 347, 359, 367, 379, 389, 433, 443, 461, 479, 487, 521, 569, 571, 577, 593, 599, 601, 617, 631, 641, 643, 647, 659, 673, 677, 719, 769, 797

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of Lucas 4-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible in mod p).

Res:= NULL: count:= 0: p:= 0:
P:= x^4 - x^3 - x^2 - x - 1:
while count < 100 do
  p:= nextprime(p);
  if [msolve(P,p)] = [] then
    Res:= Res, p; count:= count+1;
  fi
od:
Res; # Robert Israel, Mar 13 2024

t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 200}];Prime[Flatten[Position[t, 0]]]

from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A106283_gen(): # generator of terms
    p = 2
    while True:
        if len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
            yield p
        p = nextprime(p)
A106283_list = list(islice(A106283_gen(),20)) # Chai Wah Wu, Mar 14 2024

Name corrected by Robert Israel, Mar 13 2024

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

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A106280 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 4 distinct zeros.

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A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.

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