I modified this sawtooth-wave 555 osc and removed the ground at the 4k resistor, replaced it with an inverting opamp. The opamp produces a negative voltage and sinks the current, that previously went into ground after the 4k resistor. Applying 0-5V to the inverting input causes the 555 oscillate in different frequencies, where 5V gives the highest frequency and 0V the lowest.
The resistors on the inverting input and the feedback seem not to influence the sink but varying R1-R3 does. Via trial and error I set the values to somewhat "reasonable", that 0-5V produce a good range of frquencies although I have no clue how to calculate the values.
So, if I am using 0-5V at the inv. input, what formular could I use for the resistors to calculate the frequencies the 555 is doing at each voltage?