login
A301759
Semiperiods of the Fibonacci sequence mod n.
1
1, 3, 4, 6, 10, 12, 8, 12, 12, 30, 10, 24, 14, 24, 40, 24, 18, 12, 18, 60, 16, 30, 24, 24, 50, 42, 36, 48, 14, 120, 30, 48, 40, 18, 80, 24, 38, 18, 56, 60, 20, 48, 44, 30, 120, 24, 16, 24, 56, 150, 72, 84, 54, 36, 20, 48, 72, 42, 58, 120, 30, 30, 48, 96, 70, 120, 68, 36, 48
OFFSET
1,2
COMMENTS
It is the period of the Fibonacci sequence up to sign.
LINKS
Tom Harris, Notes on the Pisano Semiperiod, Dec. 2017.
David Singerman and James Strudwick, Petrie polygons, Fibonacci sequences and Farey maps, Ars Mathematica Contemporanea 10, 2 (2016), 349-357.
David Singerman and James Strudwick, The Farey Maps Modulo N, arXiv:1803.08851 [math.GR], 2018. See p. 6.
EXAMPLE
For n = 7 we get 1,0,1,1,2,3,5,1,-1,0,-1,-1,... so a(7) = 8.
MATHEMATICA
Table[NestWhile[# + 1 &, 1, ! (Mod[Fibonacci[#], n] == 0 && With[{f = Mod[Fibonacci[# + 1], n]}, f == 1 || f == n - 1]) &], {n, 69}] (* Jan Mangaldan, Sep 12 2021 *)
PROG
(PARI) a(n) = if (n==1, 1, for(k=1, oo, if (((fibonacci(k) % n) == 0) && (((fibonacci(k+1) % n) == 1) || ((fibonacci(k+1) % n) == n-1)), return (k))));
CROSSREFS
Cf. A000045 (Fibonacci numbers), A001175 (Pisano periods).
Sequence in context: A047296 A374901 A368059 * A322457 A137951 A082694
KEYWORD
nonn
AUTHOR
Michel Marcus, Mar 26 2018
STATUS
approved