This shape is only possible using the left hand thumb technique.
It gives you the the possibility of hammering on to a straight
F major to get extra twiddleability in the keys of C or F.
By moving the shape up the neck however you get some other " no 3rd " chords.
At the 2nd fret you get a spanish sounding discordant F# addb9 ( no 3rd ) - first on the sound file.
At the 3rd fret you get a nice ringy G5 ( G no 3rd )
At the 5th fret you get A7 ( no 3rd )
At the 6th fret you get Bb6 ( no 3rd )
The 7th fret gives an inversion of Gmaj7. ( Gmaj7/B )
The 8th fret gets a C5 ( C no 3rd ).
I have put some of these chords together on the soundfile ending with the Fsus2 to F major movement