Delete line if next line starts with pattern

You could try something like this:

$ sed -e '$!N;/^>.*\n>/D' -e 'P;D' file
>seq1
actgatgac
>seq2
ctgacgtca
>seq4
gtagctagt
>seq5
tgacatgca

That is

• maintain a two line buffer with $!N ... P;D
• look for a pattern that starts with > and has another > after the newline
• delete up to the newline