On the distribution of a certain family of Fibonacci type sequences
On the distribution of a certain family of Fibonacci type sequences
P. Bundschuh, Gy. Darvasi
Taking the Fibonacci sequence $G_0=1, G_1 = b \in \{ 1,3,5\}$
and $G_{n+1}= 3\cdot G_n + G_{n-1} \, (n\ge 1)$ with an integer
$2\le m\n$ , we get a purely periodic sequence $\{G_n (\mod m)\}$. Consider
any shortest full period and form a frequency block $B_m\n ^m$ to consist of
the frequency values of the residue $d$ when $d$ runs through the complete
residue system modulo $m$. The purpose of this paper is to show that such
frequency blocks can nearly always be produced by repetition of some multiple
of their first few elements a certain number of times. Theorems 3,4 and 5
contains our main results where we show when this repetition does occur, what
elements will be repeated, what is the repetition number and how to calculate
the value of the multiple.
Mathematics Subject Classification:
11B39