Can You Find the Sequence?

The sequence can be obtained by adding $+2, +10, +2, +10$ , etc. to each term in turn.

So we need to increase $a_n$ by $12$ for $n$ increasing by $2$ . $6n$ will accomplish that.

Next we need to deviate from $+6, +6$ , etc. down and then up, and down and up. If we deviate by $2$ down and then $2$ up, that will give us $4$ change in total, that is instead of $+6,+6$ , we will have $+6-4, +6+4$ , or $+2, +10$ as we want.

To get $2$ down $2$ up pattern, we can use $2\times cos(\pi n)$ .

Starting with $n=1$ our sequence at this point is: $4, 14, 16$ etc. So we need to subtract $2$ to get the numbers in the original sequence.

The answer is $6n+2cos(\pi n)-2$