Mulatu Numbers as Products of Three Generalized Lucas Numbers

Let 𝑀 𝑘 be the 𝑘-th Mulatu number. Let 𝑟, 𝑠 be non-zero integers with 𝑟 ≥ 1 and 𝑠 ∈ {−1, 1}, let {𝑈 𝑛 } 𝑛≥0 be the generalized Lucas sequence and {𝑉 𝑛 } 𝑛≥0 its companion given respectively by 𝑈 𝑛+2 = 𝑟𝑈 𝑛+1 + 𝑠𝑈 𝑛 and 𝑉 𝑛+2 = 𝑟𝑉 𝑛+1 + 𝑠𝑉 𝑛 , with 𝑈 0 = 0, 𝑈 1 = 1, 𝑉 0 = 2, 𝑉 1 = 𝑟. In this paper, we give effective bounds for the solutions of the following Diophantine equations 𝑀 𝑘 = 𝑈 𝓁 𝑈 𝑚 𝑈 𝑛 and 𝑀 𝑘 = 𝑉 𝓁 𝑉 𝑚 𝑉 𝑛 , where 𝓁, 𝑚, 𝑛 and 𝑘 are nonnegative integers and 𝓁 ≤ 𝑚 ≤ 𝑛. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell, Balancing sequences and their companions respectively.