How to create a delayed clock pulse, then return to regular clock cycles
The blue trace is one output pulse every 2.5 input pulses, not one every three.  This would take a lot of gating to decode.
For 1-out-of-3, consider the CD4017 Johnson counter.  If you OR gate or diode-OR together outputs 1, 4, and 7, that is the blue trace.  Output 8 inhibits counting and addresses a mux that switches from the output of the OR gate to the red signal (the 4017 clock input).  Not elegant, but functional.