I was working on another way of expressing these triangles when I realized there is a kind of mistake in your calculations.

When we are doing subtraction, we may find we subtract a larger number from a smaller number. This gives us a negative value with real values.

I would love to express what I have in neatly typed text but Markdown really sucks for Math. No wonder you used images for all of the notation. Mod 9 subtraction works properly by finding some number that makes the other number sum to 9. So, -8 is 1. -5 is 4.

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