Continous running light with fading trail
If you look at the surrounding methods, you'll see that 'fadeToBlackBy' will give a fading effect on anything you feed in. You'll want to locate method 'beatsin16' and then edit it yourself so that it doesn't bounce back but restarts the counter at the first LED. (hint: phase_offset)

You'll find the code snippet you want to copy in lib8tion.h

/// beatsin16 generates a 16-bit sine wave at a given BPM,
///           that oscillates within a given range.
LIB8STATIC uint16_t beatsin16( accum88 beats_per_minute, uint16_t lowest = 0, uint16_t highest = 65535,
                               uint32_t timebase = 0, uint16_t phase_offset = 0)
{
    uint16_t beat = beat16( beats_per_minute, timebase);
    uint16_t beatsin = (sin16( beat + phase_offset) + 32768);
    uint16_t rangewidth = highest - lowest;
    uint16_t scaledbeat = scale16( beatsin, rangewidth);
    uint16_t result = lowest + scaledbeat;
    return result;
}