The Delannoy numbers and the figurate numbers for n-dimensional cross polytopes are doubly-recursive sequences that satisfy the same recursion, and indeed have similar formulae, differing by one parameter. Further varying that parameter, we discover an infinite collection of doubly recursive sequences that satisfy the same recursion. These sequences enumerate certain types of lattice paths using the steps (1, 0), (0, 1), and (1,1) and certain types of words on three letters. We use these combinatorial interpretations to prove relationships among the sequences.