For your exact stated problem, you can use the keyboard shortcut of Alt-E-I-S
and press "Enter" to accept the dialog's choices. Select the last couple cells, at least, so Excel can work out your pattern of adding 2 to each previous series element, then hit the key combo.
This uses a feature that's been in Excel as long as I have used it. Not precisely "Flash Fill" or even "Fill" but those seem to use it, so same thing more or less.
Alt-H-F-I
will access the "Fill" feature in the "Home" menu (oops, sorry, the "Ribbon's Home Tab" — gosh, there's no menu-ing system there, nothing to see there at all). Then you can select a choice, though "Series" as the obvious choice gets you exactly to where the first shortcut did and more directly. Flash Fill is one of its choices.
Alt-A-F-F
accesses Flash Fill itself, directly. But it is of no use here.
Ctrl-E
, Ctrl-D
, and Ctrl-R
all do NOT do what you ask for. They can, supposedly, but it would require you to add a row for Excel to consider as the example. And... then fail anyway, usually, if the example is not complicated enough to make clear what you want.
What I cannot find and do not know, is how to exactly duplicate the mouse's achievement using the keyboard. Even the usually useless "Extend" mode does not seem to hook into it any way I've found today.