Iva,
If you only have nineteen weeks then the situation is much better, each player has 19 possible opponents and it is possible to arrange that all possible pairs of players play together exactly three times each.
(T A E K) (L N O G) (M P F I) (S B H J) (Q R C D)
(T B F L) (M O P H) (N Q G J) (A C I K) (R S D E)
(T C G M) (N P Q I) (O R H K) (B D J L) (S A E F)
(T D H N) (O Q R J) (P S I L) (C E K M) (A B F G)
(T E I O) (P R S K) (Q A J M) (D F L N) (B C G H)
(T F J P) (Q S A L) (R B K N) (E G M O) (C D H I)
(T G K Q) (R A B M) (S C L O) (F H N P) (D E I J)
(T H L R) (S B C N) (A D M P) (G I O Q) (E F J K)
(T I M S) (A C D O) (B E N Q) (H J P R) (F G K L)
(T J N A) (B D E P) (C F O R) (I K Q S) (G H L M)
(T K O B) (C E F Q) (D G P S) (J L R A) (H I M N)
(T L P C) (D F G R) (E H Q A) (K M S B) (I J N O)
(T M Q D) (E G H S) (F I R B) (L N A C) (J K O P)
(T N R E) (F H I A) (G J S C) (M O B D) (K L P Q)
(T O S F) (G I J B) (H K A D) (N P C E) (L M Q R)
(T P A G) (H J K C) (I L B E) (O Q D F) (M N R S)
(T Q B H) (I K L D) (J M C F) (P R E G) (N O S A)
(T R C I) (J L M E) (K N D G) (Q S F H) (O P A B)
(T S D J) (K M N F) (L O E H) (R A G I) (P Q B C)
Unless there is a reason for one player (T) to always be in the first foursome then it would be best to randomize the order of play within rounds to avoid this. For the 20th week you could use any arrangement of the players into foursomes, however you could make use of the scores from first 19 rounds and have a play-off for prizes.
Hope that helps,
Ian.