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I didn't find anything interesting along this line though.What about graphs represented by the Sierpinski matrix itself?

I think I'm starting to understand what Gandalf meant when he said Math World mentions a broader context for why binary logic can be used in the construction of the Sierpinski triangle.

Namely the Lucas correspondence theorem which states that given two numbers written in a prime base, $$n=n_mp^m \cdots n_1p^1 n_0p^0\space\space\space(0\le n_i\le p)$$ $$k=k_mp^m \cdots k_1p^1 k_0p^0\space\space\space(0\le k_i\le p)$$ We can get their binomial coefficient modulo that prime by performing binomial coefficients digit-wise and multiplying the results.

We can generalize from even/odd to other moduli: If we treat the rows produced by these combinatorial functions as arrays of bits, what sequence of numbers do the bits represent?

There's a variety of ways to interpret this question, but here's one assortment: The first, second, and fourth sequences are versions of each other, tautologically described in OEIS as A001317.

The difference is that polygon vertices are here rendered as points. The exponential identity for the Pascal matrix is not difficult to understand based on the series definition of the exponential function: $$e^x=\frac \frac \frac \frac \frac \frac \cdots$$ You could work out the matrix arithmetic by hand, or you could do this: $$\begin \left( \begin \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & 2 & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & 3 & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 4 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 5 & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & 6 & \cdot \\ \end \right) & \left( \begin \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 1 2 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & 2 3 & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & 3 4 & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 4 5 & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 5 6 & \cdot & \cdot \\ \end \right) & \left( \begin \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 1 2 3 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & 2 3 4 & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & 3 4 5 & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 4 5 6 & \cdot & \cdot & \cdot \\ \end \right) \\ \left( \begin \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 1 2 3 4 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & 2 3 4 5 & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & 3 4 5 6 & \cdot & \cdot & \cdot & \cdot \\ \end \right) & \left( \begin \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 1 2 3 4 5 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & 2 3 4 5 6 & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end \right) & \left( \begin \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ 1 2 3 4 5 6 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end \right) \\ \end$$ These are the first 6 powers of the subdiagonal matrix.