I have been working recently with a tensor that satisfies 

$A_{ijkl}=A_{i+b,j+b,k+b,l+b}$  $\forall$  $i,j,k,l$ $\in$ Z 

 where all indices are meant to be integers (also b with $b\geq 0$). Because of this symmetry, it is said that one can just fix one of the values of the indices, say $i=0,1,..,b-1$. Let's say that the $supp(A)=(2N+1)^{4}$ with $|i|,|j|,|k|,|l|\leq N$, where $supp(A)$ represents the support of the tensor. Then, it is argued that the total number of independent elements is given by:

$ (b-1)(2N+1)^{3}$ 

as, according to the first equation, the rest of elements can be derived by the symmetry. I am struggling to see this, as for example, considering the case of a matrix $C_{i+b,j+b}=C_{i,j}$ it is clear to me how can one do this. However, for the $A$ tensor above I am having difficulties to see this and how one could in principle recover all the missing elements of the set (the $(2N+1)^{4} -  (b-1)(2N+1)^{3}$ remaining elements )

Thanks !!