# Problem 321: Swapping Counters
A horizontal row comprising of 2n + 1 squares has n red counters placed
at one end and n blue counters at the other end, being separated by a
single empty square in the centre. For example, when n = 3. A counter
can move from one square to the next (slide) or can jump over another
counter (hop) as long as the square next to that counter is unoccupied.
Let M(n) represent the minimum number of moves/actions to completely
reverse the positions of the coloured counters; that is, move all the
red counters to the right and all the blue counters to the left. It can
be verified M(3) = 15, which also happens to be a triangle number. If we
create a sequence based on the values of n for which M(n) is a triangle
number then the first five terms would be: 1, 3, 10, 22, and 63, and
their sum would be 99. Find the sum of the first forty terms of this
sequence.