Capacity equalities in 1-dimensional (d,k)-constrained systems

In this paper, we consider the problem of determining when the capacities of distinct 1-dimensional (d,k)-constrained systems can be equal. If we let C(d,k) denote the capacity of a (d, k)-constrained system, then it is known that C(d,2d)=C(d+1,3d+1), and C(d,2d+1)=C(d+1,∞). Repeated application of these two identities also yields the chain of equalities C(1,2)=C(2,4)=C(3,7)=C(4,∞). We show that these are the only equalities possible among the capacities of (d,k)-constrained systems.