In , Deveci defined the Pell-Fibonacci sequence as follows:
P − F (n + 4) = 3P − F (n + 3) − 3P − F (n + 1) − P − F (n)
for n ≥ 0 with initial constants P − F (0) = P − F (1) = P − F (2) = 0,P − F (3) = 1. Also, he derived the permanental and determinantal representations of the Pell-Fibonacci numbers and he obtained miscellaneous properties of the Pell-Fibonacci numbers by the aid of the generating function and the generating matrix of the Pell-Fibonacci sequence. The linear recurrence sequences appear in modern research in many fields from mathematics, physics, computer, architecture to nature and art; see, for example, [2, 4, 13, 18]. In this paper, we obtain the cyclic groups which are produced by generating matrix of the Pell-Fibonacci sequence when read modulo m. Furthermore, we research the Pell-Fibonacci sequence modulo m, and then we derive the relationship between the order the cyclic groups obtained and the periods of the Pell-Fibonacci sequence modulo m.
The Pell-Fibonacci sequence, Modulo, Period