Special Checkerboard $-$ Part $3$ of $3$: Guess an analytical solution

You will now turn these checkerboards into two$-$tone checkerboards, where the white squares are untouched, but some of the dark squares will now be red instead of black. $($Any colour other than white is considered dark for this exercise$)$

$1$

is a checkerboard with $1$ black square.

Draw a separate $2$

from $1$

by adding only red and white squares.

Draw a separate $3$

from $2$

by adding only black and white squares.

Continue drawing separate $4$

and $5$

$,$ each time alternating between the colours red and black. You are asked to draw separate checkerboards because this will help you see the pattern much better.

Notice that $$

$=$ number of black squares in $$

$+$ number of red squares in $$

$.$

Look at the checkerboards you have just drawn, and express each of $1$

$,2$

$,3$

$,4$

$,5$

as a sum of two numbers using the colour coding of the drawings. Based on these values, guess a non$-$recursive formula for $$

$.$ Explain your answer

Note that there is more than one way to draw the two$-$tone checkerboards gradually as described in this question. However, only one pattern will give you an obvious formula. If you are not finding this formula, try building the checkerboards differently.