The t − Fibonacci sequences in some finite groups with center Z ( G ) = G ′

We consider finitely presented groups \(Gₘₙ\) as follows: \[ Gₘₙ= x, y {xᵐ}={yⁿ}=1, {[x, y]ˣ}=[x, y], {[x, y]ʸ}=[x, y] ~~~~~m, n≥ 2. \] In this paper, we first study the groups \(Gₘₙ\). Then by using the properties of \(Gₘₘ\) and \(t-\)Fibonacci sequences in finitely generated groups, we show that the period of \(t-\)Fibonacci sequences in \(Gₘₘ\) are a multiple of \(K(t, m)\). In particular for \(t ≥ 3\) and \(p=2\), we prove \({{LEN}ₜ({{G}ₚₚ})}= 2K(t,p)\).