The formula 1+2+3+…+n = k2 holds for only certain values of n and k.
The first solution n=1 k=1 is trivial.
The next solution is n=8 k=6. The following pictures show polyominoes of length 1-8 tiled inside a 6 x 6 square, one with L polyominoes, and the other with W polyominoes:
For more information on the fascinating world of polyominoes, see:
And don’t even get me started on the r-pentomino from Conway’s Game of Life.