The formula 1+2+3+. . .+n = 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:

The next solution is n=49 k=35. In August 2011, Berend van der Zwaag found this solution of L's:

Can anyone find a W for this case?

Rodolpho Kurchan has shown that for any value of n, these L or W polyominoes can tile a rectangle. The following pictures illustrate his method: