I've noticed that strumming from an F to an E chord, or even just the note itself, produces a characteristic flamenco sound, if the root is E. What are the other notes to this scale?
That sound you're hearing known as the Flamenco Mode, or the Major Phrygian mode.
The traditional Phrygian mode is the third mode of the major scale - essentially starting the major scale on the third degree, so E Phrygian would be: E - F - G - A - B - C - D - E
The traditional Phrygian mode is used in Flamenco, too, but to turn this into Major Phrygian (aka the Flamenco mode), just raise the third degree, so: E - F - G♯ - A - B - C - D - E
There is another variation that also raises the seventh degree: E - F - G♯ - A - B - C - D♯ - E
I should note, though, that Flamenco is a whole genre of music that is not just based off one scale. Using this mode, you can make music that sounds kind of Flamenco-like, but it is just one small ingredient in creating real Flamenco music.
The Simple Answer
The F to E transition marks the last two chords in an Andalusian Cadence (i - VII - VI - V, in your case Amin - G - F - E), which is a very prominent characteristic in Flamenco music. To think about Flamencos in a simple way, you can think about playing in a harmonic minor, which follows the pattern WS - HS - WS - WS - WS - HS - Aug 2nd - HS. You're in the key of A minor, so your harmonic minor would look like:
The G natural comes from the fact that the Andalusian scale is calculated from the natural minor. Because you are going for a major chord (indicated with V instead of v), the G# would arise as an accidental from your fifth step
The More Complicated Answer
To that point, there is such a thing called Flamenco Mode, which is the Major Phrygian mode.
In the (non-major) Phrygian mode,
you follow the pattern of HS - WS - WS - WS - HS - WH - WH. To contrast, a Major Phrygian mode would have you raise the third (and seventh, as necessary). You would be playing in E Major Phrygian:
EDIT: As a little side note, you'll notice that if you don't sharp the seventh in your Major Phrygian scale, you'll end up with something that resembles a harmonic minor scale that starts from an E instead of an A. Something is afoot! :)
You’re certainly looking for the Andalusian Cadence:
If you’re strumming or picking triad-patterns with the chords of the Phrygian cadence Am-G-F-E and improvise on the notes mi-fa-si-la, you will have the typical Flamenco sound.