t[3]
should be 5*t/3/2]
(and it is).
This is not different from the following scenario.
In[130]:= rr = RSolveValue[{t[1] == 2, t[n + 1] == 2*t[n]}, t[n], n]
(* Out[130]= 2^n *)
In[131]:= (rr /. n -> 3/2) == (2*rr /. n -> 1/2)
(* Out[131]= True *)
That is to say, the recurrence might be satisfied on a larger domain than is implied by the equatins that define it.