Here’s how to make a half double crochet (hdc), in the UK, it’s called a half treble crochet (htr).
The suggested turning chain for hdc, is ch2:
Opinions differ as to whether or not this counts as a stitch. I treat it as one, but I think the general consensus is that you don’t, just like with single crochet.
If you’re starting with a base chain, make the first hdc in the third chain from the hook.
In short: yarn over, insert hook, pull up a loop, yarn over, pull through 3 loops. Repeat!
This is a really neat stitch. I’m disappointed with myself that I haven’t used it more – I mostly use it to make gradual transitions between double crochet and single crochet. However, a fabric made up entirely of half double crochet looks really neat. I’m going to have to play with it more!