If you are on a budget you can use discrete NPN transistors or ICs with open collector (or open drain outputs) that can be scraped from old transistor radios, television sets, old printers, and other outdated electronic devices.
Discrete NPN transistors
The maximum emitter current, Ie, must be observed
Small signal transistors, like BC 547B or 2N2222 can be used, but they can only drive one of the RGB LEDs as the emitter current, Ie, will be 60 mA in your circuit and their limit is typically 100 mA. I have shown a transistor driving two in the diagram below.
Power/driver transistors, like BD 135 (1.0 A), with their much higher maximum emitter current can drive many more RGB LEDs, 16 (1.0 A/0.06 A) for BD 135.
I far as I can tell the RGB LEDs you are using are common cathode (where the "arrow" is pointing), hence the diagram above. The operating current is 20 mA and the forward voltage drops at this current are 2.0 V, 3.2 V and 3.2 V for red, green and blue, respectively.
Other values: R4 is in the kiloohm range, e.g. 3.3 kohm. One resistor is used for each internal LED as this makes for more uniform light and also accounts for the difference in forward voltage drop for red and blue/green. Vcca
is the supply voltage to the CPU and can be different from the 5 V for LEDs.
Computing the current limiting resistors
For green and blue (R2 and R3): as the current is 20 mA through the diode the same current flows through the resistor. If the voltage drop over the driver (transistor) is assumed to be 0 V then the voltage drop over the resistor is 5 V - 3.2 V = 1.8 V. We now know the current and voltage for the resistor and can use Ohm's low to find the value of the resistor:
$$ U = R3 \cdot I \implies R3 = \frac{U}{I} = \frac{1.8\ V}{0.02\ A} = 90\ \Omega $$
For red (R1):
$$ R1 = \frac{U}{I} = \frac{5.0\ V - 2.0\ V}{0.02\ A} = \frac{3.0\ V}{0.02\ A} = 150\ \Omega $$
Standard values of resistors (E24, 5%) close to these two values happens to exist (91 ohm and 150 ohm).
ICs with open collector (or open drain outputs)
The principle is the same as for the discrete transistor.
An example is the TTL 7405 (variations: 74LS05, 74HC05). The maximum current can be found in the datasheet, but most likely it can only drive one RGB LED per output. On the other hand it is more compact as there are six inverters in one IC. Some others in the TTL family (some with fewer outputs) are 7401, 74LS03, 7405, 7406, 7409, 74LS12, 74LS15, 7416, 7417, 74LS22, 74LS33, 74LS38, 74LS136, and 74LS266.
I think bus buffer/line drivers, like the 74LS244 (eight outputs) can also be used, but I have to look into it further.
References
- A good background article is "Driving LEDs with Open Drain Port Expander Outputs".
What you have guessed is correct -- if you want to run the LEDs in parallel, then you need to be able to source 600mA. If you run in series, then your power supply needs to supply at least 64V. I don't know the specifics for selecting the ideal power rail, but I would certainly shoot for >64V and put in the current limiting resistor.
A resistor on its own doesn't "drop 3.2V each" -- the voltage drop across a resistor is IxR, so the actual drop depends upon the current going through the circuit. You can calculator or measure the current going through your other group(s) of LEDs, and then pick the appropriate resistor for the mismatched group so that the current is the same.
Best Answer
If money is not of concern, you could consider a passive delay line such as these.
They claim that they can make delay lines with up to 30:1 delay to rise time. You are asking for about 13:1. Their standard products are only ~4.5:1 though so expect longer lead times and even higher prices.
Perhaps there is some other way of doing this than involving a delay...