Problem: Count the Triangles -- II

By Page Bird

Problem: Count the triangles. Extend the sequence.

Hints/Solution:
Did you get 1, 5, 13, 27 so far?
How many triangles when there are five levels? 10 levels?

 n Triangluar Numbers Up Down Total Trianlges 1 1 1 0 1 2 3 4 1 5 3 6 10 3 13 4 10 20 7 27 5 15 35 13 48 6 21 56 22 78 7 28 84 34 118 8 36 120 50 170 9 45 165 70 235 10 55 220 95 315

How do you verify your sequence?
I was able to come up with the above table after making a sketch pad drawing such as the one below and then making a table on my notebook paper. I took the hint about number of triangles that point up and down and was able to see a pattern.

I was able to make the table above to n=6 based strictly on my drawings.

My next question was could I come up with a general formula for n levels?

There is probably a more concise way to write this, however this worked for my excel explorations:

Let t(n)= the number of triangles for a given n level

Then t(n) = the number of triangles that point up plus the number of triangles that point down.

The number that point up can be found by
the summation of T(n), where T(n) is

and n begins at 1 and continues until the nth level.

The number of triangles that point down can be found by
dn= T(n -1)-dn-1
Where d is the number of triangles pointing down.

Therefore the total number of triangles can be found by
Total = T(n-1)-
dn-1 + S T(n)