I am looking for the sum of consecutive odd triangular numbers. I am trying to relate the number of $k \times k$ rhombi in an $n \times n \times n$ equilateral triangle. While I have figured out an answer to the problem as it relates to sums of triangular numbers, in particular the sum of odd triangular numbers when n is even and the sum of even triangular numbers when $n$ is odd, I am having trouble finding a closed form for the expression. For example, in a $6 \times 6 \times 6$ equilateral triangle composed of unit triangles there are $66$ total rhombi; 1 $3 \times 3$, 6 $2 \times 2$, and 15 $1 \times 1$, and we have that $T_1=1$, $T_3=6$, and $T_5=15$. $1+6+15 = 22$. And there are three types of rhombi; left, right, and vertical, so $22 \times 3$ is $66$. The same works for $n=7$ but it is the sum of $T_2 + T_4 + T_6$.
So i know how to answer the question of total rhombi, but I can not get a closed form of the summation of odd or even triangular numbers.