ILL 1974 Problem 34

Consider infinite diagrams

where all but a finite number of the integers $n_{ij} , i = 0, 1, 2, \ldots, j = 0, 1, 2, \ldots ,$ are equal to $0$ . Three elements of a diagram are called adjacent if there are integers $i$ and $j$ with $i \geq 0$ and $j \geq 0$ such that the three elements are**(i)** $n_{ij}, n_{i,j+1}, n_{i,j+2},$ or**(ii)** $n_{ij}, n_{i+1,j}, n_{i+2,j} ,$ or**(iii)** $n_{i+2,j}, n_{i+1,j+1}, n_{i,j+2}.$ An elementary operation on a diagram is an operation by which three adjacent elements $n_{ij}$ are changed into $n_{ij}'$ in such a way that $|n_{ij}-n_{ij}'|=1.$ Two diagrams are called equivalent if one of them can be changed into the other by a finite sequence of elementary operations. How many inequivalent diagrams exist?