Sum over a Triangular Grid of Arithmetic Progressions

As shown in the figure (which illustrates the case $n = 4$), an equilateral triangle $ABC$ is divided into $n$ rows of congruent small equilateral triangles, $n^2$ small triangles in all. A number is placed at every vertex of the resulting grid so that along each line of the grid, whenever the line contains at least $3$ vertices, the numbers at those vertices, taken in order, form an arithmetic progression. The three numbers placed at $A$, $B$, $C$ are pairwise distinct and their sum is $1$. Find $S_n$, the sum of the numbers at all the vertices.