Summary: The problem of optimal estimation of a linear functional \[ A_N\xi=\sum\limits_{k=0}^N a(k)\xi (k), \] that depends on unknown values of a stochastic sequence \(\{\xi (m),m\in \mathbb{Z}\}\) with stationary increments of order \(n\), by observations of the sequence at points \[ m\in \mathbb{Z}\setminus \{0,1,\dots ,N\} \] is considered. Formulas for calculating the mean square error and spectral characteristic of the optimal linear estimator of the above functional are derived in the case where the spectral density is known. In the case where the spectral density is not known, but a set of admissible spectral densities is given, the minimax-robust approach is applied to the problem of optimal estimation of the linear functional. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for a given set of admissible spectral densities.