The Skeleton Tower Task

The Skeleton Tower (A rather Halloweeny name, don't you think?) is a tower built of cubes in a pyramid fashion, with 4 stair steps extending from the center of the tower. In a tower that is 6 cubes high, there are 66 cubes altogether.

A tower 3 cubes high contains 15 cubes.

A tower 4 cubes high contains 28 cubes.

A tower 5 cubes high contains 45 cubes. 

The following table helped me in determining the number of cubes for each level. The highlighted portion was completed using mathematical equations only, without actually counting the cubes in a pyramid. 

Height	Cubes in all	Looking for Patterns
10	190	9 x 4 = 36 +1=37             *10x 19 = 190 153+37=190
9	153	8 x 4= 32 +1 = 33           *9 x 17= 153 33+ 120= 153
8	120	7 x 4=28 +1=29              *8 x 15= 120 91+29= 120
7	91	6 x 4=24 +1=25               *7 x 13= 91 66+25= 91
6	66	66-6=60          66-45=21 (the height x 4 +1) 60/4=15
5	45	45-5=40          45-28=17 (the height x 4 +1) 40/4=10
4	28	28-4=24          28-15=13 (the height x 4 +1) 24/4=6
3	15	15-3=12         15-6= 9 (the height x 4 +1) 12/4=3
2	6	6-2=4 4/4=1

The question now is whether or not there can be an equation to find the number of cubes in the pyramid at N cubes high. 

I noticed the pattern with an asterisk next to it in the table above. Obviously, 6 x 11 is 66. Initially, I thought the multiplication was just counting up odd numbers. Then I realized that 6 x 2 = 12, 7 x 2= 14, 9 x 2= 18, etc. So I wondered if I took N, multiplied it by 2, subtracted 1, then multiplied it by itself....would I get a correct answer? 

That would be [(N x 2)-1] x N.

Let's see if it is true if used for N = 4: 

[(4 x 2) - 1] x 4= [(8)-1] x 4= 7 x 4 = 28 (It works here, yay!) 

Now N= 10

[(10 x 2)-1] x 10= [(20)-1] x 10 = 19 x 10= 190 (Yay again!)

Using the table, I was able to identify patterns that would help formulate an equation to solve the problem. Oddly enough, the pattern that helped the most was just a side note when I noticed that all of the numbers in the not highlighted section were multiples of the height in cubes.