Effective Bounds for Narayana Numbers that Admit Zeckendorf Partition in k Terms: Explicit Results for k ∈ {3, 4}

The mathematician Zeckendorf proved that every positive integer has a unique decomposition into a sum of distinct and nonconsecutive Fibonacci numbers, known as the Zeckendorf partition. Let be the Narayana’s cows sequence given by N ₀ = N ₁ = N ₂ = 1 and N _{m +3} = N _{m +2} + N _m ,  for  m ≥ 0. A term of this sequence is called a Narayana number. In this paper, we applied the linear forms in logarithms to the Zeckendorf decomposition of the Narayana cow’s sequence . We give effective computable bounds for the decomposition. As applications, we explicitly determine all the Narayana numbers that admit a Zeckendorf partition into exactly three terms or exactly four terms.