Electronic – Design a Counter With an Arbitrary Sequence

You can create a 3-input, 3-output circuit that maps one state in the count sequence to the next. Then feed the 3 Q outputs of the FF's into this combinatorial circuit, and connect the output of the combinatorial circuit to the 3 D inputs of the FF's. Connect all 3 of the FF's clock lines together, and then whenever an active edge comes along the flip flops act like a 3 bit register, and just load up the next value.

Think of your special counter as a state machine. Then assign the state the coded value of the count sequence that you want. In this case the states would be as following with the next state showing.

State    Next State
  3  ->   1
  1  ->   4
  4  ->   7
  7  ->   2
  2  ->   3

Each state can be encoded into three binary bits so your design will require three D type flipflops. You need to make up a set of three karnough maps, one for each flipflop that, shows the next bit value for the flop flop (D input) based upon the three current state (Q outputs).

Use the k-maps to simplify the logic down to the minimal required. Finally you can code the minimal logic up in a series of AND gates driving OR gates into each FF D-input.

Here I show the k-map for the lowest ordered bit of the "counter" to get you started with the idea.