A self-similarity structure generated by kingʹs walk Original Research Article

For all non-negative integers i,j let w(i,j) denote the number of all paths in the plane from (0,0) to (i,j) with steps (1,0), (0,1), and (1,1). The numbers w(i,j) are known as the Delannoy numbers. They were studied by several authors. Let p be an odd prime and let w̄(i,j) denote the remainders of w(i,j) when divided by p where 0⩽w̄(i,j)