A104217 Period of Perrin (0,2,3,2,5,5,..., A001608) sequence mod n.

1, 7, 13, 14, 24, 91, 48, 28, 39, 168, 120, 182, 183, 336, 312, 56, 288, 273, 180, 168, 624, 840, 22, 364, 120, 1281, 117, 336, 871, 2184, 993, 112, 1560, 2016, 48, 546, 1368, 1260, 2379, 168, 1723, 4368, 231, 840, 312, 154, 2257, 728, 336, 840, 3744, 2562

Offset: 1

Author

Anthony C Robin, Mar 14 2005

Keywords

• nonn

Comments

Analogy to A001175, Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n AND to A046738 for Perrin sequence, where a(n)=a(n-2)+a(n-3)

It appears that the n such that n-1 divides a(n) is the set of primes of the form x^2+23*y^2 (A033217). The discriminant of the characteristic polynomial of the Perrin sequence is -23. - T. D. Noe, Feb 23 2007

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A001175, A046738 and Perrin sequence A001608.

Programs

Mathematica

Table[a={0,2,3}; a=a0=Mod[a, n]; k=0; While[k++; s=a[[2]]+a[[1]]; a=RotateLeft[a]; a[[ -1]]=Mod[s,n]; a!=a0]; k, {n,100}] (* T. D. Noe, Oct 10 2006 *)

Python

from math import lcm
from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A104217(n):
    if n < 4:
        return (1,7,13)[n-1]
    f = factorint(n).items()
    if len(f) > 1:
        return lcm(*(A104217(a**b) for a,b in f))
    else:
        k,x = 1, (0,2,3)
        while x != (3,0,2):
            k += 1
            x = (x[1], x[2], (x[0]+x[1]) % n)
        return k # Chai Wah Wu, Apr 25 2025

Extensions

More terms from T. D. Noe, Oct 10 2006