Arduino sketch to cycle an RGB LED through the colour spectrum.

rgb_spectrum.c

const int redPin = 11;
const int greenPin = 10;
const int bluePin = 9;

void setup() {
  // Start off with the LED off.
  setColourRgb(0,0,0);
}

void loop() {
  unsigned int rgbColour[3];

  // Start off with red.
  rgbColour[0] = 255;
  rgbColour[1] = 0;
  rgbColour[2] = 0;  

  // Choose the colours to increment and decrement.
  for (int decColour = 0; decColour < 3; decColour += 1) {
    int incColour = decColour == 2 ? 0 : decColour + 1;

    // cross-fade the two colours.
    for(int i = 0; i < 255; i += 1) {
      rgbColour[decColour] -= 1;
      rgbColour[incColour] += 1;
      
      setColourRgb(rgbColour[0], rgbColour[1], rgbColour[2]);
      delay(5);
    }
  }
}

void setColourRgb(unsigned int red, unsigned int green, unsigned int blue) {
  analogWrite(redPin, red);
  analogWrite(greenPin, green);
  analogWrite(bluePin, blue);
 }

for states :
255, 0, 0
255, 255, 0
0, 255, 0
0, 255, 255
0, 0, 255
255, 0, 255
255, 0, 0

void loop() {
  unsigned int rgbColour[3];

  // Start off with red.
  rgbColour[0] = 255;
  rgbColour[1] = 0; 
  rgbColour[2] = 0;  
  int decColour =0;
  int incColour =1;

void loop() {
			if (255<=rgbColour[incColour]) {
				if (0>=rgbColour[decColour]) {
					decColour++;	decColour %= 3;
					incColour++;	incColour %= 3;
				} else 	// if (0==rgbColour[decColour]) {
					rgbColour[decColour] -= 15;  // --
			} else	// if (255==rgbColour[incColour]) {
				rgbColour[incColour] += 15;  // ++
      
        setColourRgb(rgbColour[0], rgbColour[1], rgbColour[2]);
        delay(5);
    }
}

Here is some code that address the issue pointed by @michalmonday (eg: colors such as (255, 255, 0), (255, 0, 255), (0, 255, 255) never reached). Additionally, it's a non-blocking version. Unlike original code, it does not keep 100% brightness anymore (eg: at some point 2 leds are fully on, sometimes only one...).

const int redPin = 11;
const int greenPin = 10;
const int bluePin = 9;
unsigned char rgbColor[3] = { 255, 0, 0 }; //start with red
int state = 0;

unsigned char colors[6]= { 1, 0, 2, 1, 0, 2 }; //green, red, blue
unsigned char directions[6]= { 1, -1, 1, -1, 1, -1 }; //up, down

void setup() {
	// Start off with the LED off.
	setColourRgb(0,0,0);
}

void loop() {
	int color = colors[state];
	rgbColor[color] += directions[state];
	if (rgbColor[color] == 0 || rgbColor[color] == 255) {
		state = (state + 1) % 6; //next state
	}
	
	setColourRgb(rgbColor[0], rgbColor[1], rgbColor[2]);
	delay(3);
}

void setColourRgb(unsigned char red, unsigned char green, unsigned char blue) {
	analogWrite(redPin, red);
	analogWrite(greenPin, green);
	analogWrite(bluePin, blue);
}