Does this type of graph have a name? The type of graph that arises when drawing Pascal's triangle, see the link here. Only depends on one integer (the depth) and it is sort of like a binary tree after identifying certain vertices (of course it is no longer a tree then). I think they must have a name somewhere since they are fairly natural things to consider... but I can't find a name anywhere using Google.
 A: It's a triangular grid graph, which is a special case of a lattice graph.
Technically it also contains the horizontal connections, which are not part of your example, but they could as well be included. 
EDIT: Without them it is actually just a square lattice grid as pointed out by @Joffan.
Citing Weisstein, Eric W. "Triangular Grid Graph." From MathWorld--A Wolfram Web Resource:

The triangular grid graph $T_n$ is the lattice graph obtained by interpreting the order-$(n+1)$ triangular grid as a graph, with the intersection of grid lines being the vertices and the line segments between vertices being the edges. Equivalently, it is the graph on vertices $(i,j,k)$ with $i,j,k$ being nonnegative integers summing to $n$ such that vertices are adjacent if the sum of absolute differences of the coordinates of two vertices is $2$ (West 2000, p. 391).
The graph bandwidth of $T_n$ is $n+1$ (West 2000, p. 392).
$T_n$ is also the hexagonal king graph of order $n$, i.e., the connectivity graph of possible moves of a king chess piece on a hexagonal chessboard.
