Abstract. Nanotubical graphs are obtained by wrapping hexagonal grid into a cylinder, and then possibly closing the tube with patches. In the paper we determine the number of vertices at distance d from a particular vertex in an open (k,l)-nanotubical graph. This number depends mainly on the circumference and not on the type of the nanotubical graph. In particular, for d>2k-1 it is 2(k+l) in an infinite open nanotube. These calculations are used in a subsequent papers to determine asymptotics for various distance-based invariants for nanotubical graphs.