A104217 - OEIS

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

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OFFSET

1,2

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

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 *)

PROG

(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

CROSSREFS

Cf. A001175, A046738 and Perrin sequence A001608.

Sequence in context: A275201 A120100 A308525 * A269163 A260482 A328254

Adjacent sequences: A104214 A104215 A104216 * A104218 A104219 A104220

KEYWORD

nonn

AUTHOR

Anthony C Robin, Mar 14 2005

EXTENSIONS

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

STATUS

approved