Triangle+Counting+Challenge

If anybody comes up with anything important on the Triangle question, feel free to post it on this page. Hopefully we can collaborate and get an answer!

Sorry about the squished table. The **Bolded** numbers represent the number of rows (n), the Blue represents the side length (x) of a right side up triangle, the Red represents the side length of an upside down triangle, and the normal numbers are the number of this kind of triangle that occur when you have **n** rows.

From looking at the table we can see that the pattern is 1, 3, 6 etc. which is just 1+2+3.... For right side up triangles, the pattern starts when x=n. For upside down triangles, the pattern starts when 2x=n.

When we think of side length x, the number of x lengthed triangles =the number of 2x lengthed triangles.


 * || **1** || **2** || **3** || **4** || **5** || **6** || **7** || **8** || **9** || **10** || **11** || **12** || **13** || **14** ||
 * 1 || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 || 55 || 66 || 78 || 91 || 105 ||
 * 2 ||  || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 || 55 || 66 || 78 || 91 ||
 * 3 ||  ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 || 55 || 66 || 78 ||
 * 4 ||  ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 || 55 || 66 ||
 * 5 ||  ||   ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 || 55 ||
 * 6 ||  ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 ||
 * 7 ||  ||   ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 ||
 * 8 ||  ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 ||
 * 9 ||  ||   ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 ||
 * 10 ||  ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 ||
 * 11 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 || 10 ||
 * 12 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 ||
 * 13 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 3 ||
 * 14 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 ||
 * 1 ||  || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 || 55 || 66 || 78 || 91 ||
 * 2 ||  ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 || 55 || 66 ||
 * 3 ||  ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 || 36 || 45 ||
 * 4 ||  ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 || 21 || 28 ||
 * 5 ||  ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 || 10 || 15 ||
 * 6 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 3 || 6 ||
 * 7 ||  ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 ||
 * //Total// || //1// || //5// || //13// || //27// || //48// || //78// || //118// || //170// || //235// || //315// || //411// || //525// || //658// || //812// ||
 * //Total// || //1// || //5// || //13// || //27// || //48// || //78// || //118// || //170// || //235// || //315// || //411// || //525// || //658// || //812// ||

According to Stewart, this means we have two different functions (probably one for right side up triangles, and one for upside down triangles)
 * Number of Rows (n) || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 ||
 * Total || 1 || 5 || 13 || 27 || 48 || 78 || 118 || 170 || 235 || 315 || 411 || 525 || 658 || 812 ||
 * 1st Difference || 4 || 8 || 14 || 21 || 30 || 40 || 52 || 65 || 80 || 96 || 114 || 133 || 154 ||
 * 2nd Difference ||  || 4 || 6 || 7 || 9 || 10 || 12 || 13 || 15 || 16 || 18 || 19 || 21 ||
 * Every Second Difference (7-4), (9-6) etc. || 3 || 3 || 3 || 3 || 3 || 3 || 3 || 3 || 3 || 3 ||  ||
 * Every Second Difference (7-4), (9-6) etc. || 3 || 3 || 3 || 3 || 3 || 3 || 3 || 3 || 3 || 3 ||  ||