Hi tonywilson824,
You’ll want to set the base current high enough to bring the transistor into the saturation region, where the Vce voltage is brought down to a minimum value (for the least internal power dissipation and heat generation) and where the maximum collector current is above what your circuit will actually draw (I believe you stated above that it would be about 250mA). You can set the base current using a series resistor to any value less than what the Arduino can source, which is about 20mA, but you won’t need nearly this much, as seen below.
To do this you can look at Figure 4 on page 4 of the MPSA29 datasheet.
Figure 4. CollectorSaturationRegion
Figure 4 shows a few things of note. First, it shows that for any collector current, there is a minimum base current required to get to that collector current. For instance, to reach 250mA of collector current, you will need a minimum of close to 20uA of base current. But at 20uA base current, the voltage drop across Vce will be about 2V. This means that less voltage is available to drop across your LEDs and it will dissipate quite a bit of power. Since 250mA will be passing through the transistor and the voltage drop will be 2V, the transistor will have to dissipate 500mW (2V x 0.25A = 0.500W) of power as heat.
If you increase the base current to, say, about 800uA, the Vce voltage drop reduces to about 0.9V for a 250mA collector current. In addition to leaving more of the voltage to drop across your LEDs, this reduces the transistor power dissipation to only about 225mW.
The area on the curves where the line for a given collector current transitions from a more vertical slope to a more horizontal slope is where the transistor starts to become “saturated”. This is the region you want to be in when turning on and off a load because it minimizes lost power in the transistor and minimizes the voltage dropped across the collector-emitter region.
So, it looks like setting the base current to at least 500nA should do the job, and there is no harm in doubling that to 1mA to get really good saturation. The Arduino can easily handle supplying this current. Your formula was right, since there are two diode drops in a Darlington transistor. Looking at Figure 2, it shows that the actual Vbe will be roughly between 1.5V and 1.6V.
Figure 2.“ON” Voltages
To be sure you have enough base current, one should choose the higher Vbe voltage, which means a lower base resistor value to allow enough current.
So, to get about 1mA of base current, choose a base resistor by subtracting the voltage drop across the Vbe pins (about 1.6V) from your Arduino I/O pin voltage of about 5V and dividing that by 1mA.
Rbase = (Vi/o - Vbe) / I base = (5V - 1.6V) / 0.001A = 3400Ω
This does not need to be very precise, so a 5% resistor should be fine. The nearest common 5% value to 3400Ω (or 3.4kΩ) is 3.3kΩ, so that’s a good value to try.