technically, there is no proof given here of that most important last part that (n,k)%3==0 for k!=0,n and for n==3^d, though it is not hard to deduce, yet not that obvious.

For tuples addition is defined as concatenation, so sum(< iterator of tuples >, ()) simply concatenates the tuples generated by the zip object. We have to specify the starting value (second argument) as an empty tuple here, because by default sum() will start by adding 0 + < tuple > which will throw the TypeError exception.

seems like this solution will stuck in dead ends