This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A106276 #11 Feb 16 2025 08:32:57
%S A106276 1,0,0,1,2,1,1,1,0,1,0,0,1,1,3,3,0,1,0,0,1,1,1,0,0,1,3,1,1,0,1,1,0,1,
%T A106276 1,1,0,3,1,1,0,0,0,1,1,3,1,0,1,0,1,1,1,0,3,1,3,1,1,1,1,1,1,3,0,0,0,1,
%U A106276 1,1,0,1,0,1,0,0,0,3,3,1,3,3,1,0,1,0,0,1,1,0,0,1,0,1,3,1,0,0,1,1,1,1,1,1,1
%N A106276 Number of distinct zeros of x^3-x^2-x-1 mod prime(n).
%C A106276 This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step recursions, A000073 and A001644. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -44 = -4*11. The primes p yielding 3 distinct zeros, A106279, correspond to the periods of the sequences A000073(k) mod p and A001644(k) mod p having length less than p. The Lucas 3-step sequence mod p has two additional primes p for which the period is less than p: 2 and 11, which are factors of the discriminant -44. For p=11, the Fibonacci 3-step sequence mod p has a period of p(p-1).
%H A106276 J.-P. Serre, <a href="https://doi.org/10.1090/S0273-0979-03-00992-3">On a theorem of Jordan</a>, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
%H A106276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>.
%t A106276 Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]
%Y A106276 Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106293 (period of the Lucas 3-step sequences mod prime(n)), A106282 (prime moduli for which the polynomial is irreducible).
%K A106276 nonn
%O A106276 1,5
%A A106276 _T. D. Noe_, May 02 2005