In this paper, we represent the admissible solutions of the system of second-order rational difference equations given below in terms of Lucas and Fibonacci sequences: {eqnarray*} {split} xₙ₊₁={Lₘ₊₂+Lₘ₊₁yₙ₋₁}{Lₘ₊₃+Lₘ₊₂yₙ₋₁}, yₙ₊₁={Lₘ₊₂+Lₘ₊₁zₙ₋₁}{Lₘ₊₃+Lₘ₊₂zₙ₋₁},\\ zₙ₊₁={Lₘ₊₂+Lₘ₊₁wₙ₋₁}{Lₘ₊₃+Lₘ₊₂wₙ₋₁}, wₙ₊₁={Lₘ₊₂+Lₘ₊₁xₙ₋₁}{Lₘ₊₃+Lₘ₊₂xₙ₋₁}. {split} {eqnarray*} where n₀, ₘ=-∞+∞ is Lucas sequence and the initial conditions x₋₁, x₀, y₋₁, y₀, z₋₁, z₀, w₋₁, w₀ are arbitrary real numbers such that v₋ᵢ≠-Lₘ₊₃Lₘ₊₂, where v₋ᵢ=x₋ᵢ,y₋ᵢ,z₋ᵢ,w₋ᵢ, $i=0,1$ and m.
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Berkal et al. (2023) studied this question.
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