using one 555 timer to control the pulse of another
Your circuit looks okay.  I redrew the circuit in CircuitLab so I could simulate it.  Although you showed the circuit using two 555's, and I followed that, you can get both timers in one package as a 556.

Your timing diagram shows two pulses, but with 6 seconds, and a 1 second on, 1/2 off pulse from the astable, you are going to get three.  So I left it at that.

I verified the values of the resistors and capacitors you chose (first substituting µF for F) using two calculators, one for the monostable and another for the astable). The timing for the monostable was right at 6 seconds, and for the astable at 1.47 seconds with a 67% duty cycle (0.98 seconds on, 0.49 seconds off).

I then changed the time for the monostable to 5.5 seconds instead of 6, since four time periods of 1.5 seconds equals 6 seconds, and depending on how accurate your timing is, you might get the beginning of another pulse from the astable right at the 6 second mark.  At least I did in the simulation.  So I used a 50K resistor for R1 rather than the 55K one.  Obviously you can change these values with the calculators.

When I was done, I used the CircuitLab simulator.  This is what I got:



As you can see, there are three pulses within the six seconds as expected. The first pule is longer, because as Dave Tweed points out, the timing capacitor of the second timer is charging all the way from 0 volts to 2/3 Vcc, while for the subsequent pulses, it's only charging from 1/3 Vcc to 2/3 Vcc. 

So the circuit should work as it is drawn, as far as I can tell.