Does anyone know if it is possible to get brown with RGB LEDs?
Sounds like all you need is an amber LED run at very low levels - like 5mA through it - that'll come out a dull ochre.
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If you have power supplies with fine enough granularity, then yes. Any color which can be encoded RGB (e.g., 24-bit X11 colors) can be produced with an RGB LED. If you have a digital control for your power supplies (say, if they're PWMs set through a microcontroller) you could even feed the 24-bit code directly. A decent basic brown is #803000 (hex).
Would it be possible to use some resistors to control the current through the LED because I am trying to keep this project as cheap and simple as possible?
That's always possible, provided you just want a fixed, never-changing color. Given the RGB code, you can calculate the resistors you need pretty trivially. You should provide more detail in your question text if you want people to be able to give you the inforation you need.
Is there a formula for calculating the resistor values? I was just going to figure it out with trial and error. And thanks for the advice I 'll be more specific in the future!
Yes. If you do a Web search for "LED calculator" (or even search for that here on I'bles), you'll find several that can do the job for you. Once you have a set of "balanced" resistors (i.e., where all the LEDs are equal intensity, just scale up the ones you need to be dimmer to get the colors you want.