Fade an LED on an Arduino Uno

// Fade: a breathing LED on pin 9, from the timer, not from the CPU.
//
// Pin 9 is OC1A, timer 1's compare output. analogWrite() puts timer 1 into
// 8-bit phase-correct PWM with a 64 prescaler, which is 16 MHz / (64 x 510) =
// 490 Hz, and after that the hardware holds the duty cycle on its own: the
// sketch only changes OCR1A thirty times a second and sleeps in between.
//
// Wire an LED and a 220 ohm resistor from pin 9 to GND.
//
// The table is one cycle of (1 - cos)/2, which is what makes the light look
// like it is breathing rather than ramping: the eye reads brightness roughly
// logarithmically, so a linear ramp looks like it rushes the top end.

const int LED_PIN = 9;
const int STEPS = 64;
const int MS_PER_STEP = 30;

const uint8_t breath[STEPS] = {
    0,   1,   2,   5,  10,  15,  21,  28,
   37,  46,  56,  66,  77,  89, 101, 113,
  125, 137, 149, 161, 173, 184, 194, 204,
  213, 222, 229, 235, 240, 245, 248, 249,
  250, 249, 248, 245, 240, 235, 229, 222,
  213, 204, 194, 184, 173, 161, 149, 137,
  125, 113, 101,  89,  77,  66,  56,  46,
   37,  28,  21,  15,  10,   5,   2,   1,
};

void setup() {
  pinMode(LED_PIN, OUTPUT);
}

void loop() {
  for (int step = 0; step < STEPS; step++) {
    analogWrite(LED_PIN, breath[step]);
    delay(MS_PER_STEP);
  }
}