Investigate!

For the patterns of dots below, draw the next pattern in the sequence. Then give a recursive definition and a closed formula for the number of dots in the $n$th pattern.

$A sequence of three dot patterns, starting with a single dot (labeled n = 0), then the same dot with four new dots extending into an X (labeled n = 1) and finally the same pattern but with four new dots extending onto the limbs of the X (labeled n = 2).$

$A sequence of three dot patterns, labeled n = 0, n = 1, and n = 2, from left to right. The first pattern contains two dots, on above the other. The second pattern can be viewed as each of the dots from the first pattern splitting into 3, forming two triangles (so six dots total). The third patter again takes each dot from the second pattern and splits them into three, arranged as triangles (so 18 dots all together).$

$A sequence of four dot patters, labeled n = 1 through n = 4. The first pattern is a single dot. The second adds a row of two dots below the single dot (so three dots in a triangle). The third pattern adds a row of three dots, creating six dots arranged in a triangle. Finally we add four dots below the previous six, still forming a triangle, this time of 10 dots.$