You've posted some great and useful information as well. The patterns using this method are fascinatingly complex and beautiful. I encourage you to play!
Here is the code, identical to attached above:
'LED Strip APA-102
'A faster way to signal the LEDs
' We will pwmout the clock with one timer, and pwmout the data line with another timer.
' Phase shifts between the frequencies should result in interesting patterns
' albeit with poor...
With only a few lines of code you can make an arbitrarily long string of APA102s dance!
There are several fine posts describing driving the APA102 with PICAXE:
https://picaxeforum.co.uk/threads/apa102-rgb-led-strip-examples.30680/...