notation:: #A is the number of factors of A, B(e,n)={x $\in$A|d(e,x)≦n}, and S(e,n)={x $\in$A|d(e,x)=n}
Then, I want to know that #B(e,n) of $\mathbb{Z}^k$.
Where $\mathbb{Z}^k$ is equipped with the word length metric associated to the standard generating set.
I can understand the case of k=1,2 ,since I can imagine the grapth of $\mathbb{Z}$ and $\mathbb{Z}^2$.
I think that each point contained in S(e,n) connect the adjacent points except for the case of contained in B(e,n-1) ,but I cannot count the duplicative points. Please teach me how to think this problem.