

A088551


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


2



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
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OFFSET

2,2


COMMENTS

If pi(n) is the nth 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 semicolon.


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}] (* JeanFrançois 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



