1,2

This sequence differs from the corresponding Fibonacci sequence (A106302) at n=1 and 5 because these correspond to the primes 2 and 11, which are the prime factors of -44, the discriminant of the characteristic polynomial x^3-x^2-x-1. We have a(n) < prime(n) for the primes 2, 11 and A106279.

For a prime p, the period depends on the zeros of x^3-x^2-x-1 mod p. If there are 3 zeros, then the period is < p. If there are no zeros, then the period is p^2+p+1 or a simple fraction of p^2+p+1. Also note that the period can be prime, as for p=3, 5, 31, 59, 71, 89, 97, 157, 223. When the period is prime, the orbits have a simple structure. [From T. D. Noe (noe(AT)sspectra.com), Sep 18 2008]

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

n=3; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 60}]

Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106279 (primes p such that x^3-x^2-x-1 mod p has 3 distinct zeros), A106302 (period of Fibonacci 3-step sequence mod prime(n)).

Sequence in context: A097443 A100589 A111489 this_sequence A101649 A063305 A166143

Adjacent sequences: A106291 A106292 A106293 this_sequence A106295 A106296 A106297

nonn

T. D. Noe (noe(AT)sspectra.com), May 02 2005