On partial sum of Tribonacci numbers (Q1751354)

Summary: We study the sum \(s_t^{(k, r)} = \sum_{i = 0}^t T_{k i + r}\) of \(k\) step apart Tribonacci numbers for any \(1 \leq r \leq k\). We prove that \(s_t^{(k, r)}\) satisfies certain Tribonacci rule \(s_t^{(k, r)} = a_k s_{t - 1}^{(k, r)} + b_k s_{t - 2}^{(k, r)} + s_{t - 3}^{(k, r)} + \lambda\) with integers \(a_k, b_k, c_k\), and \(\lambda\).