A088551 Fibonacci winding number: the number of 'mod n' operations in one cycle of the Fibonacci sequence modulo n.

1, 3, 2, 8, 11, 7, 4, 11, 28, 3, 9, 12, 23, 19, 9, 16, 11, 7, 28, 5, 12, 23, 9, 48, 40, 35, 19, 4, 59, 12, 19, 15, 16, 39, 9, 36, 6, 27, 28, 19, 19, 43, 11, 59, 23, 15, 9, 55, 148, 35, 38, 52, 35, 6, 21, 31, 16, 26, 57, 28, 12, 21, 43, 68, 51, 67, 14, 19, 119, 32, 7, 72, 112, 99, 5, 33

Offset: 2

Author

R C Johnson (bob.johnson(AT)dur.ac.uk), Nov 19 2003

Keywords

• easy
• nice
• nonn

Comments

If pi(n) is the n-th Pisano number (A001175) then a(n) is usually about pi(n)/2 - and in any case a(n) > pi(n)/4.

Examples

			a(8)=4 because one cycle of the Fibonacci numbers modulo 8 is 0, 1, 1, 2, 3, 5; 0, 5, 5; 2, 7; 1; - including 4 'mod 8' operations, each marked with a semi-colon.
		

Links

• T. D. Noe, Table of n, a(n) for n=2..10000
• R. C. Johnson, Fibonacci Numbers and Resources.
• M. Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Int. Seq. 14 (2011) # 11.9.1.

Crossrefs

Cf. A001175, A015134, A214300.

Programs

Mathematica

(* pp = Pisano period = A001175 *) pp[1] = 1;
pp[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k + 1], n] == 1, Return[k]]];
a[n_] := Sum[Mod[Fibonacci[k], n], {k, 1, pp[n]}]/n;
Table[a[n], {n, 2, 77}] (* Jean-François Alcover, Sep 05 2017 *)

Formula

n*a(n) = sum{k=1..A001175(n)} fibonacci(k) mod n. [Mircea Merca, Jan 03 2011]

Extensions

More terms from T. D. Noe

Edited by Ray Chandler, Oct 26 2006