%I #16 Jun 29 2013 10:54:37
%S 1,1,2,3,4,8,12,22,45,86,155,357,675
%N Number of runs of strictly increasing numbers of 2 X 2 squares in the list of partitions of an n X n square lattice into squares, considering only the list of parts, where partition sorting order is ascending with larger squares taking higher precedence.
%C The sequence was derived from the documents in the Links section. The documents are first specified in the Links section of A034295.
%C a(n) = row length of A226948(n).
%H Jon E. Schoenfield, <a href="https://oeis.org/A034295/a034295.txt">Table of solutions for n <= 12</a>
%H Alois P. Heinz, <a href="https://oeis.org/A034295/a034295_1.txt">More ways to divide an 11 X 11 square into sub-squares</a>
%H Alois P. Heinz, <a href="https://oeis.org/A034295/a034295_2.txt">List of different ways to divide a 13 X 13 square into sub-squares</a>
%e For n = 3, the partitions are:
%e Square side 1 2 3
%e 9 0 0
%e 5 1 0
%e 0 0 1
%e So a(3) = 2 as there are 2 runs of 2 X 2 squares: (0,1) from the first 2 partitions and (0) from the third.
%Y Cf. A034295, A226948.
%K nonn,more
%O 1,3
%A _Christopher Hunt Gribble_, Jun 23 2013