A086797 -id:A086797 - 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

A056788 a(n) = n^n + (n-1)^(n-1).

Original entry on oeis.org

2, 5, 31, 283, 3381, 49781, 870199, 17600759, 404197705, 10387420489, 295311670611, 9201412118867, 311791207040509, 11414881932150269, 449005897206417391, 18884637964090410991, 845687005960046315793, 40173648337182874339601, 2017766063735610126699403

Offset: 1

Walter Nissen, Aug 20 2000

nonn

For even n > 1, the absolute value of the discriminant of the polynomial x^n+x-1. [Corrected by Artur Jasinski, May 07 2010]

The largest known prime in this sequence is a(4) = 283.

			a(3) = 2^2 + 3^3 = 4 + 27 = 31.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see equation (6.7).

Pierre Letouzey, Generalized Hofstadter functions G, H and beyond: numeration systems and discrepancy, arXiv:2502.12615 [cs.DM], 2025. See p. 18.
Walter Nissen, post on np ( n ) = n^n + (n+1)^(n+1), on home page "Up for the count!". (Updated Oct 02 2012)

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

Cf. A056187, A056790, A192397 (smallest & largest prime factor of a(n), records of the latter), A217435 = bigomega(a(n)).

Join[{2}, Table[n^n+(n-1)^(n-1), {n, 2, 20}]] (* T. D. Noe, Aug 13 2004 *)
Join[{2},Total/@Partition[Table[n^n,{n,20}],2,1]] (* Harvey P. Dale, Jun 26 2017 *)

A056788(n)=n^n+(n-1)^(n-1)  \\ M. F. Hasler, Oct 02 2012

Minor corrections by M. F. Hasler, Oct 02 2012

A238194 Conjectured numbers n for which n^n + (-1)^n (n-1)^(n-1) is not squarefree.

Original entry on oeis.org

130, 257, 487, 528, 815, 897, 1176, 1225, 1320, 1373, 1430, 2029, 2050, 2084, 2198, 2247, 2526, 2608, 2895, 2936, 2958, 3166, 3679, 3849, 3909, 3950, 4237, 4319, 4598, 4647, 4723, 4795, 5472, 5487, 5620, 5669, 5948, 6030, 6317, 6358, 6588, 6677, 6936, 7101

Offset: 1

T. D. Noe, Feb 24 2014

The first case (130) yields a number divisible by 83^2. The next 5 terms yield numbers divisible by 59^2. Boyd et al. are not completely certain about the other 994 numbers up to 1000. They conjecture that 0.9934466... of numbers n^n + (-1)^n (n-1)^(n-1) are squarefree.
Boyd et al. tested the values n <= 1000 for divisibility by the squares of the first 10^4 primes. To extend the sequence, I tested the divisibility of n <= 200000 by the squares of the first 10^5 primes. - Giovanni Resta, Feb 24 2014
The heuristic chance that Resta's list is incomplete is just over 1%. This drops to 0.07% with testing to the millionth prime. - Charles R Greathouse IV, Feb 25 2014

David W. Boyd, Greg Martin, and Mark Thom, Squarefree values of trinomial discriminants, arXiv 1402.5148 [math.NT], 2014.
Chandrashekhar Khare, Alfio Fabio La Rosa, and Gabor Wiese, Splitting fields of X^n - X - 1 (particularly for n = 5), prime decomposition and modular forms, Univ. du Luxembourg (2022).
Giovanni Resta, Terms < 200000 and corresponding square divisors

Cf. A086797 (n^n + (-1)^n (n-1)^(n-1) with signs).

is(n)=!issquarefree(n^n + (-1)^n*(n-1)^(n-1)) \\ Charles R Greathouse IV, Feb 25 2014

a(7)-a(44) from Giovanni Resta, Feb 24 2014

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

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Dec 21 2005

sign

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).

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.

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).

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

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

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

cp's OEIS Frontend

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

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A056788 a(n) = n^n + (n-1)^(n-1).

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A238194 Conjectured numbers n for which n^n + (-1)^n (n-1)^(n-1) is not squarefree.

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

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