Journal of Integer Sequences, Vol. 19 (2016), Article 16.2.6 |

Department of Mathematics

Shimane University

Matsue, Shimane 690-8504

Japan

**Abstract:**

We consider a *generalized Fibonacci sequence* ( *G*_{n} ) by
and
*G*_{n} = *G*_{n-1} + *G*_{n-2} for any integer *n*.
Let *p* be a prime number and let *d*(*p*) be the smallest positive integer *n* which satisfies
.
In this article, we introduce equivalence relations for the set of generalized Fibonacci sequences.
One of the equivalence relations is defined as follows.
We write
if there exist integers *m* and *n* satisfying
.
We prove the following: if
*p* ≡ 2 (mod 5),
then the number of equivalence classes
satisfying
for any integer *n* is
(*p*+1)/*d*(*p*)-1.
If
*p* ≡ ± 1 (mod 5), then
the number is
(*p*-1)/*d*(*p*)+1.
Our results are refinements of a theorem given by Kôzaki and Nakahara in 1999.
They proved that there exists a generalized Fibonacci sequence ( *G*_{n} )such that
for any
if and only if one of the following three conditions holds:
(1) *p* = 5; (2)
*p* ≡ ± 1 (mod 5);
(3) *p* ≡ 2 (mod 5)
and *d*(*p*)<*p*+1.

Received November 7 2015; revised versions received January 18 2016; January 20 2016; January 25 2016.
Published in *Journal of Integer Sequences*, February 5 2016.

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