patterns for nth term rules

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?

was quite a challenge for a while

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...

some regular versions were found

what commonalities are there?
and why?


it seemed that a pattern to fit the rule 6n + 5 might be impossible

for quite a long time

then someone had the novel idea of having 'open top' patterns

these are all the 4th patterns in
6n + 3
6n + 2
and 6n + 5

now for 6n - 1 ...