Linear regression models are checked by a lack-of-fit (LOF) test to be sure that the model is at least approximatively true. In many practical cases data are sampled sequentially. Such a situation appears in industrial production when goods are produced one after the other. So it is of some interest to check the regression model sequentially. This can be done by recursive least squares residuals. A sequential LOF test can be based on the recursive residual partial sum process. In this paper we state the limit of the partial sum process of a triangular array of recursive residuals given a constant regression model when the number of observations goes to infinity. Furthermore, we state the corresponding limit process for local alternatives. For specific alternatives designs are determined dominating other designs in respect of power of the sequential LOF test described above. In this context a result is given in which $e^{-1}$ plays a crucial role.