OFFSET
1,1
COMMENTS
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014), # 14.8.5.
A. Elsenhans and J. Jahnel, The Fibonacci sequence modulo p^2 -- An investigation by computer for p < 10^14, (2004).
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Wikipedia, Pisano period.
FORMULA
a(n) = A001175(prime(n)). - Jonathan Sondow, Dec 09 2017
a(n) = (3 - L(p))/2 * (p - L(p)) / A296240(n) for n >= 4, where p = prime(n) and L(p) = Legendre(p|5); so a(n) <= p-1 if p == +- 1 mod 5, and a(n) <= 2*p+2 if p == +- 2 mod 5. See Wall's Theorems 6 and 7. - Jonathan Sondow, Dec 10 2017
MAPLE
a:= proc(n) option remember; local F, k, p;
F:=[1, 1]; p:=ithprime(n);
for k while F<>[0, 1] do
F:=[F[2], irem(F[1]+F[2], p)]
od: k
end:
seq(a(n), n=1..70); # Alois P. Heinz, Oct 16 2015
MATHEMATICA
Table[p=Prime[n]; a={1, 0}; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[2]]=s; a!=a0]; k, {n, 100}] (* T. D. Noe, Jun 12 2006 *)
PROG
(PARI) for(n=1, 100, s=1; while(sum(i=n, n+s, abs(fibonacci(i)%prime(n)-fibonacci(i+s)%prime(n)))+sum(i=n+1, n+1+s, abs(fibonacci(i)%prime(n)-fibonacci(i+s)%prime(n)))>0, s++); print1(s, ", "))
(Python)
from itertools import count
from sympy import prime
def A060305(n):
x, p = (1, 1), prime(n)
for k in count(1):
if x == (0, 1):
return k
x = (x[1], (x[0]+x[1]) % p) # Chai Wah Wu, May 31 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Louis Mello (mellols(AT)aol.com), Mar 26 2001
EXTENSIONS
Corrected by Benoit Cloitre, Jun 04 2002
Name clarified by Jonathan Sondow, Dec 09 2017
STATUS
approved