Lets start from the beginning. Let say we can't change C, so our capacitor is 2200pf or 2.2 nf. And the request is a 3Khz oscillator running off of a 6V supply.
The formula for the chip as a single supply is derived directly from the time constant.
so my time constant, which is 1/3Khz =0.000333333 seconds
So my R for the time constant, for a 2.2nF capacitor, at 0.000333333 seconds, would have to be calculated as: time/capacitance or 0.000333333/0.0000000022 = 151515.151515152 ohms. Since my target is 50% duty cycle, but don't care on precision, my Rb resistance value can be derived by charge current voltage drop between the threshold and discharge pin, which would be 2/3 of a voltage drop of e R in the RC time constant, or 151515.151515152 x 0.66666666 = 101010.090909091. The best close find would be a 101K resistor for Rb. The calculation of Ra in this shortcut formula is 1% of Rb or 1.01K. Which can be simulated here
Now you say you want a +5/-5 square wave oscillator, with a positive and negative 5 volt supply. That is simple. Since our threshold trigger is 1/2 of the total wave compared to a single supply, you double the capacitance and the resistor simulated here.
Just a reminder: This is a quick and sloppy way to set up a timer. If you need precision, always use the formulas on the datasheet.
Edit: more to come. Just need to dig for my 555 Timer schematic to show you why the output is not going exactly to 0V on its low state.