Ruler is a bit confusing. Let's use scale for weighing.

With 2 pieces, we can weigh up to 4 by using 1 and 3.

1 = 1

2 = 3 - 1

3 = 3

4 = 1 + 3

With 3 pieces, we can weigh up to 11 by using 1, 3, and 7. Up to 4 are same as above.

5 = 1 + 7 - 3

6 = 7 - 1

7 = 7

8 = 1 + 7

9 = 3 + 7 - 1

10 = 3 + 7

11 = 1 + 3 + 7

Next would be 4 pieces to weigh 19. I believe it can be proved by binary system.