The name of the triangular arrangement of numbers in order

Question

What do you call a triangular arrangement of numbers in order from the top, like in this picture? I know of Pascal's triangle but this is different.

I know that the string that appears on the right-hand diagonal sequence of this triangle consists of triangle numbers, but don't have any idea what to call the string that runs through the center of this vertically, or this arrangement itself, if there is any name for this at all.

Oops, sorry, the title of my question was confusing. What I was mainly asking was the name of the arrangement of numbers like the one seen in the picture linked, not the numbers in this arrangement because they are just integers...

Answer 1

In that arrangement, all diagonal lines and all vertical lines are second-order arithmetic progressions and so are given by quadratic expressions.

For instance, the repeated differences for the center vertical line are $$ \begin{array}{llll} 1 & 5 & 13 & 25 & \\ 4 & 8 & 12 & \\ 4 & 4 & \\ 0 & \\ \end{array} $$ Newton's interpolation formula then gives us $$ 1 \binom{n-1}{0} + 4 \binom{n-1}{1} + 4 \binom{n-1}{2} $$ which simplifies to $$ 2 n^2 - 2 n + 1 $$