Why do you think that A should be independent of t? Your calculation looks mostly correct to me, except for one thing: k is supposed to be an integer, so the condition means that the greatest allowable value of k is (the greatest integer not exceeding t/3). Then the convolution (note the spelling!) is the function

At first sight, that looks like a constant times . But in fact depends on t and is not a constant. (However, it only changes value at integer values of t/3, and is constant in the intervening intervals.)