The Root raised cosine filter output depends on previous input

I'm trying the RRC filter, and I see a problem that the output of the filter y(t) is actually depends on the input x(t) and x(t-1)

For example, I design a system that create new input every sixth samples:

At t = 0, the output is y1(t)

At t = 1, ... 5, the output is y2(t) which is different from y1(t), although the input is identical

At t = 6, the output is y3(t)

At t = 7, ... 11, the output is y4(t) which is again different from y3(t), although the input is identical

As I understand a filter should not have this property, can someone please explain if it is correct?