Three series LEDs on 5 V?
I know that when I use only one LED, it has a potential difference of 2V and need 5 - 20mA or it will burn, so:


Good. Then you should know that if you have three of those LEDs in series that you will need 6 V. Since you only have 5 V that explains the dim LEDs.


  R=V/I => 5-2/0.015 = 200ohm &lt;210ohm


Not quite. It should be \$ R = \frac {V}{I} = \frac {5 - 3 \times 2}{0.015} = \frac {-1}{0.015} = -66.6 \; \Omega \$. Since negative resistors don't exist this is a sign that something is wrong.

The most you can light on 5 V is two LEDs but even that depends on the colour (which in turn depends on the doping added to the LED material).



Figure 1. Forward current vs voltage for a range of typical LEDs. Note the variation between colours. Source: LEDnique.com.

Be careful with your micro-controller output pin. There will be a maximum current that it can safely provide and the voltage will droop a bit - maybe 0.5 V or so at that current. There is also a maximum that all the outputs can provide at one time because the current has to be sourced or sunk from the Vss or Vdd pin. The datasheets will contain all this information.