1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411.
Background:
This comes from a question of how many total triangles are there in a large triangle whose base equals n of the smallest possible triangles (there are the maximum number of small triangles in the large triangle such that there is no overlap and no white spaces in the large triangle). The series above are the answer for each of the first 11 sizes of the large triangle (where the base equals fits 1 small triangle, 2, small triangles, 3 small triangles ... 11 small triangles). So if the large triangle's base only fits 1 of the smallest triangles the total triangles is 1. If the large triangle's base fits 2 of the smallest triangles the total is 5 (4 of the smallest plus 1 of the largest). After this you need to include all of the medium sized triangles as well.
How far I've gotten:
for notation it is best to consider triangles by size (1x1 being the smallest possible).
- (1x1) --> 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21
- (2x2) --> 0, 1, 2, 4, 6, 8, _10, 12, 14, 16, 18
- (3x3) --> 0, 0, 1, 2, 3, 5, __7, _9, 11, 13, 15
- (4x4) --> 0, 0, 0, 1, 2, 3, __4, _6, _8, 10, 12
- (5x5) --> 0, 0, 0, 0, 1, 2, __3, _4, _5, _7, _9
- (6x6) --> 0, 0, 0, 0, 0, 1, __2, _3, _4, _5, _6
- ...
- (11x11)-->0, 0, 0, 0, 0, 0, _0, _0, _0, _0, _1
The pattern should be clear. Each column is the number of new triangles for each size as the base of the large triangle increases in size in relation to the smallest triangle. So the total triangles for a given base size is the total of each column plus those previous to it.
I'm looking for an equation that will give me the total of any triangle whose base is size n.
Any thoughts?
