I am looking for a 555/556 circuit that runs a 5 minute delay then produces a single 3 second negative pulse (pull-down). Would I use two separate timers and if so how would I interface the two?

 

How do you want to initiate the 5min timer? Power on? Circuit closure? pulse?

 

Circuit closure

 

This will not be easy. It requires a large resistor and capacitor and the leakage of the cap needs to be small. I would use a micro controller ( very often my advice ).

 

If you wanted Astable you could use a 470uF capacitor, with 820kohm and 15kohm resistors.

If you wanted one shot, then a 556 with 220uF and an 1200kohm resistor triggering to a 4.7uF and 1000k resistor should do it

 

I would suggest counters or a programmable timer.

The warning Russ gave about the large capacitors needing to be low leakage is important.

The CD4541 is very useful once you understand the added functionality it offers.

 

Using a different device as suggested is probably your best option but if you really want to use a 555 or 556 and avoid the capacitor leakage problem, you can use large value ceramic capacitors and multiple timing stages.

 

According to the datasheets on capacitors, this can't be done because electrolytic capacitors leak too much. It isn't true. I've done it.

There is also a CMOS version of the 555 that makes this easier. It has better input specs.
Warm up that soldering iron and get to experimenting!

 

I was thinking more on the lines of a 5 miniute timer start a 3 second timer on time out of the 5 min.

 

A CD4541 has an oscillator and a divider allowing a small reasonable cap to be used to set very long time intervals. You may even be able to get the 3 sec delay out of it too by playing with the output and reset lines.

The CD4541 is one of those oddball chips that don't quite fit any category but turn out to be damn useful.

 

The capacitor leakage problem is overstated in a major sort of way. I have used a 555's for timers that lasted for hours using large caps. You can see what is happening by putting a simple DVM across the cap and watching the voltage over time.

For an inverted signal such as the OP suggests a transistor will likely also be needed.

 

The problem is not just getting the values but getting in the right values and stable over time. As often happens an Arduino is a fairly easy solution that typically gives great accuracy and flexibility. Not very expensive when you consider price per hour of entertainment, in fact for one off projects on all but the leanest budgets is an easy decision to defend. At some point you pretty much have to learn microcontrollers.

 

Now is a good time as any to learn how to use a microcontroller.
You can solve this with a simple 8-pin chip and no other external components.