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

2,2

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.

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.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A001175, A015134, A214300.

Sequence in context: A209360 A095013 A094188 * A301903 A165660 A171634

Adjacent sequences: A088548 A088549 A088550 * A088552 A088553 A088554

Keyword

easy,nice,nonn

Author

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

Extensions

More terms from T. D. Noe

Edited by Ray Chandler, Oct 26 2006

Status

approved