A222659 - OEIS

%I #14 May 29 2013 14:57:13

%S 1,2,2,4,8,4,8,34,34,8,16,148,320,148,16,32,650,3118,3118,650,32,64,

%T 2864,30752,68480,30752,2864,64,128,12634,304618,1525558,1525558,

%U 304618,12634,128

%N Table a(m,n) read by antidiagonals, m, n >= 1, where a(m,n) is the number of divide-and-conquer partitions of an m X n rectangle into integer sub-rectangles.

%C The divide-and-conquer partition of an integer-sided rectangle is one that can be obtained by repeated bisections into adjacent integer-sided rectangles.

%C The table is symmetric: a(m,n) = a(n,m).

%e Table begins:

%e 1,      2,       4,       8,      16,     32,      64, ...

%e 2,      8,      34,     148,     650,   2864,   12634, ...

%e 4,     34,     320,    3118,   30752, 304618, 3022112, ...

%e 8,    148,    3118,   68480, 1525558, ...

%e 16,   650,   30752, 1525558, ...

%e 32,  2864,  304618, ...

%e 64, 12634, 3022112, ...

%e Not every partition (cf. A116694) into integer sub-rectangles is divide-and-conquer. For example, the following partition of a 3 X 3 rectangle into 5 sub-rectangles is not divide-and-conquer:

%e 112

%e 342

%e 355

%Y a(1,n) = a(n,1) = A000079(n-1)

%Y a(2,n) = a(n,2) = A034999(n)

%Y Cf. A116694 (all partitions).

%K tabl,nonn,more

%O 1,2

%A _Arsenii Abdrafikov_, May 29 2013