Here is a 16 player schedule. It's not quite perfect since pairs (5,16) & (6,15) meet twice as opponents, all other pairs who meet twice do so once as partners and once as opponents.

(13 15 v 14 4) (11 6 v 2 10) (12 16 v 8 5) ( 1 9 v 7 3)

(16 7 v 13 11) ( 1 6 v 12 15) ( 8 4 v 3 14) ( 5 2 v 10 9)

( 8 16 v 6 4) ( 2 15 v 3 5) ( 9 11 v 1 13) (14 10 v 12 7)

( 8 12 v 2 13) ( 9 6 v 15 14) (10 16 v 1 3) ( 5 4 v 11 7)

( 1 2 v 4 13) ( 3 15 v 12 11) ( 6 8 v 9 7) (10 5 v 16 14)

( 7 2 v 15 16) (11 1 v 8 14) ( 4 9 v 12 10) ( 5 13 v 3 6)

(11 16 v 9 12) ( 6 5 v 4 1) (10 15 v 13 8) ( 7 14 v 2 3)