A106294 Period of the Lucas 3-step sequence A001644 mod prime(n).

1, 13, 31, 48, 10, 168, 96, 360, 553, 140, 331, 469, 560, 308, 46, 52, 3541, 1860, 1519, 5113, 5328, 3120, 287, 8011, 3169, 680, 51, 1272, 990, 12883, 5376, 5720, 18907, 3864, 7400, 2850, 8269, 162, 9296, 2494, 32221, 10981, 36673, 4656, 3234, 198, 5565

Offset

1,2

Comments

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, Sep 18 2008]

Links

Table of n, a(n) for n=1..47.

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

Formula

a(n) = A106293(prime(n)).

Mathematica

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}]

Crossrefs

Cf. A106273, A106279, A106302.

Sequence in context: A348300 A100589 A111489 * A101649 A063305 A301622

Adjacent sequences: A106291 A106292 A106293 * A106295 A106296 A106297

Keyword

nonn

Author

T. D. Noe, May 02 2005

Status

approved