formula for number triangles Hi, I have a triangle starting from $0$ and going up by one on the bottom row until there are $r$ items on the bottom row and there are $r$ rows a number is formed by adding the two numbers towards the fat end of the triangle together.

Example:
  $$x \downarrow\ $$
  $$
\begin{matrix}12\end{matrix}\\
\begin{matrix}4 & 8\end{matrix}\\
\begin{matrix}1 & 3 & 5\end{matrix}\\
\begin{matrix}0 & 1 & 2 & \boxed{3}\end{matrix} 
$$
  $$\begin{matrix}& & r - 1 \uparrow\end{matrix}$$
  $\therefore$ when $r = 4$, $x = 12$.

When $r = 8$
  $$
\begin{matrix}0\ \ & 1\ \ & 2\ \ & 3\ \ & 4\ \ & 5\ \ & 6\ \ & 7\ \ \end{matrix}\\
\begin{matrix}1\ \ & 3\ \ & 5\ \ & 7\ \ & 9\ \ & 11\ & 13\ \end{matrix}\\
\begin{matrix}4\ \ & 8\ \ & 12\ & 16\ & 20\ & 24\ \end{matrix}\\
\begin{matrix}12\ & 20\ & 28\ & 36\ & 44\ \end{matrix}\\
\begin{matrix}32\ & 48\ & 64\ & 80\ \end{matrix}\\
\begin{matrix}80\ & 112 & 144 \end{matrix}\\
\begin{matrix}192 & 256 \end{matrix}\\
\begin{matrix}448\end{matrix}\\
$$
  $\therefore x = 448$

How do I work out $x$ from $r$? What is the formula? Any help by suggesting a new viewpoint or formula(e) would be greatly appreciated. Thank you.

Consider the first two 'columns'.  Notice any patterns?
$$0 \space 1\\1 \space 3 \\4 \space 8\\ 12\space  20$$
What is the rule?
$$a_{i+1} = a_i + b_i$$
$$b_{i+1} = a_{i+1} + 2 (b_i - a_i) = 3b_i - a_i$$
Try find a formula for $a_n$.
