Enter triangle numbers
When looking at the odds for 3d6, the beginning few - 1,3,6,10 - show a familiar pattern: the triangular numbers. T(n) is known to be \(\dfrac {n(n+1)}{2}\) . But when going further in, they begin to overestimate the result, and for the final numbers give way more – e.g. for \(n_{16} =1\) it predicts \(\dfrac {16(16+1)}{2} = 136\)
One may notice, the discrepancy comes form the fact that certain combinations the formula would predict are, in fact, impossible. One such example is \(9=1+1+7\) as a d6 always is less than 7.