%I #21 Jun 04 2025 10:24:24

%S 0,0,0,1,4,15,39,88,162,283,450,691,1005,1425,1954,2626,3444,4452,

%T 5652,7094,8775,10755,13035,15676,18679,22053,25819,29967,34543,39531,

%U 44976,50878,57231,64026,71296,79026,87243,95920,105036,114590,124672,135206,146231,157684,169642,182051,194927,208298,222125,236484

%N a(n) is the number of ways to partition a square n X n into five rectangles of different dimensions, without any straight cut spanning the entire square.

%C Alternatively a(n) is the total number of distinct sets of five unordered integer duplets with distinct element composition of the form: (x,y), (p,y+q), (n-p,q), (n-p-x,n-q), (p+x,n-y-q) where elements of a duplet represent the lengths of the two sides of a rectangle, p+x < n, q+y < n and 0 < x,y,p,q < n.

%e When n = 5,the duplet (5,5) can be decomposed in the following four different ways:

%e {(1,1), (1,2), (1,4), (2,3), (3,4)},

%e {(1,1), (1,3), (2,2), (2,4), (3,3)},

%e {(1,2), (1,3), (1,4), (2,2), (3,4)},

%e {(1,3), (1,4), (2,2), (2,3), (2,4)}.

%e In each case a rectangle is surrounded by four rectangles of different dimensions. Each of the four surrounding rectangles shares part of one its sides with a side of the central rectangle (x,y) and extends to the boundary of the square in that direction.

%Y Cf. A381847.

%K nonn

%O 1,5

%A _Janaka Rodrigo_, May 22 2025