take a bite out

I came across this task in Dan Walker's 'sequences' work on the TES website (looking for number patterns) and think it's one of the best generalisations ever...

the intention is that students establish a general rule numerically
which can be established algebraically if students can multiply out brackets

but, more interestingly I think, they can be asked to consider various forms of diagrams
"what might a picture/diagram of this look like?"

some are presented here - students could be asked to show that what is left (after a bite has been taken out) can be filled with the numbers in the associated addition sum
and convince themselves (and others) that their 'gap-filling' is systematic and can continue...