I had many creative entries, via email, blog comments, and in a Ravelery thread. You created many lovely stitches, dear readers, but only one is really right.
The Tsock Tsarina figured it out first... I give thee the Tsarina Brioche:
Cast on an odd number of sts.
R1 (WS): * p1, yf sl1yo; rep from *, end p1
R2: * k1, yf sl1yo; rep from *, end k1 (when you slip the stitch you also slip the yo that goes with it, so that in the next row you have the stitch and BOTH yos to pick up with the brp.)
R3: p1, yo, * brp1, yf sl1yo; rep from *, end last rep p1 instead of yf sl1yo
R4: k1, * k1, brk1; rep from *, end last rep k1 instead of brk1
yf sl1yo (yarn forward, slip 1 purlwise, yarnover) = bring the working yarn under the needle to the front of the work, slip the next stitch purlwise, then bring the yarn over the needle (and over the slipped stitch) to the back in position to work the following stitch; this slipped stitch/ yarnover is considered 1 stitch.
brk (brioche knit - also known as a bark) = knit the stitch (that was slipped in the row before) together with its yarnover.
brp (brioche purl - also known as a burp) = purl the stitch (that was slipped in the row before) together with its yarn over
Unfortunately I am not the brilliant Yarn Jedi that the Tsarina is (This is a woman, after all, that designs leafy socks), and kept screwing up the swatch. It wasn't until Nephele posted her simplified version that I was able to actually knit the #%*^@ thing.
Over uneven number of stitches:
Row 1: Purl
Row 2: Knit
Row 3: Purl
Row 4: *K1, Knit into stitch two rows down, repeat from * to last stitch, K1
Row 5: Purl
Row 6: K1, *k1, knit into stitch 1 row down, repeat from * to last 2 stitches, k2
Repeat rows 3 through 6 for pattern.
I'll be sending out my yarny gratitude to both. And now I have to make something with this... hmmm...