it's often quite interesting to reverse a question:
here's the answer - what was the question?
in this case, given an nth term rule
what might a growing 'matchstick' pattern look like?
then someone hit upon the idea of e.g. starting with 2 and adding 6 each time
some of their patterns were not sufficiently regular...
what commonalities are there?
it seemed that a pattern to fit the rule 6n + 5 might be impossible
for quite a long time
these are all the 4th patterns in
6n + 3
6n + 2
and 6n + 5
now for 6n - 1 ...