Recap: the compositions of n can be generated by always moving one unit of the rightmost part to the left and spreading its remaining units out to the left. Example (n=5): [ 1 1 1 1 1 ] [ 1 1 1 2 ] [ 1 1 2 1 ] [ 1 1 3 ] [ 1 2 1 1 ] [ 1 2 2 ] [ 1 3 1 ] [ 1 4 ] [ 2 1 1 1 ] [ 2 1 2 ] [ 2 2 1 ] [ 2 3 ] [ 3 1 1 ] [ 3 2 ] [ 4 1 ] [ 5 ] Now slightly change the above recipe by moving that "one unit" by _two_ places. Example (n=9): [ 1 1 1 1 1 1 1 1 1 ] [ 1 1 1 1 1 1 2 1 ] [ 1 1 1 1 1 2 2 ] [ 1 1 1 1 2 2 1 ] [ 1 1 1 1 3 2 ] [ 1 1 1 2 3 1 ] [ 1 1 1 3 3 ] [ 1 1 2 3 1 1 ] [ 1 1 2 4 1 ] [ 1 1 3 4 ] [ 1 2 3 1 1 1 ] [ 1 2 3 2 1 ] [ 1 2 4 2 ] [ 1 3 4 1 ] [ 1 4 4 ] [ 2 4 1 1 1 ] [ 2 4 2 1 ] [ 2 5 2 ] [ 3 5 1 ] [ 4 5 ] The number of such compositions is given by http://oeis.org/A011972 (note that such compositions of the triangular numbers give http://oeis.org/A000110 ). Can anyone shed some light on this? This riddle has been bugging me for a long time, and I still cannot see anything pertinent. Best, jj