How to design, build and test synchronous sequential circuits using D-Flip Flops?
You want to generate this sequence right?

For that start with truth table.

 000 --&gt; 00111 
 001 --&gt; 00101
 010 --&gt; 10000 
 011 --&gt; 11011 
 100 --&gt; 00001 
 101 --&gt; 10100 
 110 --&gt; 11100 
 111 --&gt; 11111


First you make 3 bit synchronous counter using D flip-flop. Use counter outputs as input variable for logic circuit to get respective output combination. 
First draw 3 variable (outputs of counter) K-map for Z1. Find boolean equation for the same. Repeat this for Z2, Z3, Z4 and Z5.

If you need more clarification let me know.