• • A Novel Third-Order Recurrence Relation and Its Convergence to the Golden Ratio

This paper formalizes a newly discovered integer sequence defined by a unique third-order linear recurrence relation using a deterministic term-skipping mechanism. Discovered independently by Nandkishor Prajapat, this sequence utilizes the initial seed values () and operates under the recurrence rule \(a_n = 2aₙ₋₂ + aₙ₋₃\). Despite utilizing architectural frameworks that diverge completely from traditional second-order systems like the Fibonacci and Lucas sequences, we mathematically demonstrate that this sequence possesses hidden cross-connections to the Lucas numbers and structurally converges to the cosmic constant known as the Golden Ratio (φ ≈ 1.618).