# [seqfan] Re: Property of Fibonacci sequence modulo M

Richard Mathar mathar at strw.leidenuniv.nl
Wed Dec 2 20:15:32 CET 2009

```jb> Date: Wed, 02 Dec 2009 07:57:06 -0800
jb> From: Jack Brennen <jfb at brennen.net>
jb> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
jb> Subject: [seqfan]  Property of Fibonacci sequence modulo M
jb>
jb> Consider that v(M) consists of one period of the infinite repeating
jb> sequence of Fibonacci numbers taken modulo M -- the shortest sequence
jb> of numbers of length L such that v == 1, v == 1, v[L-1] == 1,
jb> and v[L] == 0, with v[n] == (v[n-2]+v[n-1])%M.
jb>
jb> So v(5) would be:
jb>    1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0
jb>
jb> Now, consider a(M) to be the number of times that the integer 1
jb> occurs in v(M).  So, for instance a(5) == 4.

In OEIS speak: the smallest row k for which A128924(k,1) = n.

I see that the k-Fibonacci sequences a(n+1)=k*a(n)+a(n-1) are in the OEIS
for k=1 (*the* Fibonacci sequence),
k=2 (A000129)
k=3 (A006190)
k=4 (A001076)
k=5 (0 followed by A052918)
k=6 (A005668)
k=7 (0 followed by A054413)
k=8 (0 followed by A041025)
k=9 (A099371)
k=10 (0 followed by A041041)
k=11 (0 followed by A049666)
k=12 (0 followed by A041061)
Are the associated Pisano period lengths in the OEIS, generalizing A001176?

These are discussed by
S. Falcon and A. Plaza in "k-Fibonacci sequences modulo m", Chaos, Solitons & Fractals 41 (10) (2009) 497-504
http://dx.doi.org/10.1016/j.chaos.2008.02.014

Richard

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