I'm trying work on my ability to break complex patterns down,
and in this case I'm trying to model the denominators of Lacsap's Fractions:
I managed to get the sequence that represents the denominators:
But now I want to convert it to a series in order to get rid of the tedious term D_n-1.
I guess I'm just not quite sure how to apply the sum identity to a parametric sequence:
The red is the answer that I've found works, but I don't know how to get to it.
When I simply substitute k, D_1, and D_k with snippets from the sequence, I get it wrong.
I think you wanted to write $$D_{n,r}=D_{n-1,r}+(n-r)$$ though you could have written $$D_{n,r}=D_{n-1,r-1}+r$$ with the combined effect that (assuming that you start from $D_{0,0}=0$) $$D_{n,r}=\sum_{i=1}^{n-r} i + \sum_{j=1}^{r} j = \frac{(n-r)(n-r+1)}{2} + \frac{r(r+1)}{2}$$ and hence your result.
Another way to look at it is to note that $D_{n,0}=D_{n,n}=\frac{n(n+1)}{2}$ and that $D_{n,r}=D_{n-r,0}+D_{r,r}$. For the numerator you have $N_{n,r}=\frac{n(n+1)}{2}$.