Q$1(10$ points$)$

Special Checkerboard $-$ Part $1$ of $3$: Recursive Definition

In this question you will be working with special checkerboards that have a square shape and where all four corners are dark, such as this one:

$360$px$-$Checkerboard$\_$pattern.svg$.$png

Define the size n of the checkerboard to be the number of dark squares on each side of the checkerboard, and call such a checkerboard $$

The checkerboard in the picture above is $3$

Define $$

$=$ to be the number of dark squares in $$

$.$ As we can see in the example above, $3$

$=13.$

Give a recursive definition of $$

$.$ This definition must include:

All the initial conditions needed to define this sequence $($do not give more or fewer than needed$)$

A recurrence relation for $$

as a function of some of the elements of the sequence that precede $$

The values of n for which this recurrence relation applies

Explain this recursive definition. This explanation must include:

an explanation of the recurrence relation. Support this explanation with drawings of $1$

$,2$

$,3$

$,4$

$.$

an explanation of how you decided how many initial conditions were needed for this recursive definition

Note that the explanations are the most important part of this question. Do not simply give answers for your mathematical formulas. Explain your reasoning!