A301759 Semiperiods of the Fibonacci sequence mod n.

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

Table of n, a(n) for n=1..69.

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: A176865 A047296 A368059 * A322457 A137951 A082694

Adjacent sequences: A301756 A301757 A301758 * A301760 A301761 A301762

Keyword

nonn

Author

Michel Marcus, Mar 26 2018

Status

approved