[seqfan] Re: three triangular numbers in a row

[David Wilson]

Something inspired by this thread:

Let T(a) denote the triangle of a. Here is a neat geometric way of 
generating
solutions to

     3 T(a) = T(b)

Start with the base solution a = 1, b = 2, that is

     3 T(1) = T(2)

We draw an equilateral triangle of side b = 2, affix a triangle of side 
a = 1
to each corner, then extend the outer edges of the side-1 triangles to 
form a
larger side-9 triangle:

            -------------------------------------
            \ c   c   c   c \ a / c   c   c   c /
             \               \ /               /
              \ c   c   c   c / c   c   c   c /
               \             / \             /
                \ c   c   c / b \ c   c   c /
                 \         /     \         /
                  \ c   c / b   b \ c   c /
                   \---------------------/
                    \ a / c   c   c \ a /
                     \ /             \ /
                      \ c   c   c   c /
                       \             /
                        \ c   c   c /
                         \         /
                          \ c   c /
                           \     /
                            \ c /
                             \ /


Knowing that the 3 a-triangles add up to the b-triangle, we can conclude 
that
the three side-5 triangles add up to the side-9 triangle, and we have a new
solution

     3 T(5) = T(9).

So can redraw the figure with a = 5 and b = 9. This will show us that 3 
side-20
triangles add up to a side-35 triangle.

In general, this shows that a solution

     3 T(a) = T(b)

implies a larger solution

     3 T(2a + b + 1) = T(3a + 2b + 2).

Note also that if you start out knowing

     3 T(5) = T(9)

the same figure can be used to conclude

     3 T(1) = T(2)

In other words, if you have a solution, you can use the figure to keep 
finding
smaller solutions until you arrive at 3 T(1) = T(2). This shows that 
solutions
found by this process are the only solutions in positive integers.