Continuing the previous question: $($which was: 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: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!

This problem is continued in the next question.$)$ Continuing the previous question:

You will now use iteration to deduce a partial solution involving $\backslash$Sigma

or $\backslash$Pi

operators for the sequence $$

:

Give the first $5$ terms of the sequence. Show and keep the intermediate expansions because they are more important than the final values for noticing a pattern $($and your grade will depend on it$).$

Guess a non$-$recursive formula which describes the sequence. The formula should include $\backslash$Sigma

or $\backslash$Pi

operators and should be as compact as possible.

The pedagogical goal of this question is not to find an analytical solution for $$

$,$ but to learn how to use iteration to notice patterns in sequences, and to write them correctly and succinctly using $\backslash$Sigma

and $\backslash$Pi

notation.

In order to do this, you must work from intermediate values instead of final values. Do distribute your operations to remove the parentheses in each term of the sequence, but do not calculate the results of additions, multiplications, and exponentiations, because if you do the pattern will disappear.

This problem is continued in the next question.