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A222659 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. 0

%I

%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

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Last modified January 20 03:25 EST 2022. Contains 350467 sequences. (Running on oeis4.)