  We consider the totally asymmetric simple exclusion process (TASEP) in
discrete time with sequential update. The joint distribution of the positions
of selected particles is expressed as a Fredholm determinant with a kernel
defining a signed determinantal point process. We focus on periodic initial
conditions where particles occupy dZ, d>=2. In the proper large time scaling
limit, the fluctuations of particle positions are described by the Airy_1
process. Interpreted as a growth model, this confirms universality of
fluctuations with flat initial conditions for a discrete set of slopes.
