Finding an equation for a pattern

Question

I am trying to figure out how to form an equation for the total number of tiles in Figure "N." I am given that these are the following patterns for the tiles/figures: 1, 1+3+1, 1+3+5+3+1, 1+3+5+7+5+3+1, 1+3+5+7+9+5+3+1. These are the first 5 figures and the patterns that arise through each of them. I am trying to figure what an equation for this would be because each time you add, you add by two and each time you subtract, you subtract by 2. However, how do I make an equation so it knows when to start decreasing by 2? Thanks!

Answer 1

$$\begin{array} &u_1&=\underbrace{1}_{1^2}&=1^2\\ u_2&=\underbrace{1+3}_{2^2}+\underbrace{1}_{1^2}&=2^2+1^2\\ u_3&=\underbrace{1+3+5}_{3^2}+\underbrace{3+1}_{2^2}&=3^2+2^2\\ u_4&=\underbrace{1+3+5+7}_{4^2}+\underbrace{5+3+1}_{3^2}&=4^2+3^2\\ \vdots\\ u_n&=\underbrace{1+3+5+\cdots+(2n-1)}_{n^2}+\underbrace{\cdots+5+3+1}_{(n-1)^2}&=n^2+(n-1)^2\\ &&=\color{red}{2n^2-2n+1} \end{array}$$

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NB:

$$\begin{align} u^n &=\sum_{r=1}^n (2r-1)^2+\sum_{r=1}^{n-1} (2(n-r)-1)^2\\ &=\sum_{r=1}^n (2r-1)^2+\sum_{r=1}^{n-1} (2r-1)^2\\ &=n^2+(n-1)^2\\ &=2n^2-2n+1 \end{align}$$