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

Author

Michel Marcus, Mar 26 2018

Keywords

• nonn

Comments

It is the period of the Fibonacci sequence up to sign.

Examples

			For n = 7 we get 1,0,1,1,2,3,5,1,-1,0,-1,-1,... so a(7) = 8.
		

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.

Crossrefs

Cf. A000045 (Fibonacci numbers), A001175 (Pisano periods).

Programs

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 *)

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))));