A384208 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.

0, 0, 0, 1, 4, 15, 39, 88, 162, 283, 450, 691, 1005, 1425, 1954, 2626, 3444, 4452, 5652, 7094, 8775, 10755, 13035, 15676, 18679, 22053, 25819, 29967, 34543, 39531, 44976, 50878, 57231, 64026, 71296, 79026, 87243, 95920, 105036, 114590, 124672, 135206, 146231, 157684, 169642, 182051, 194927, 208298, 222125, 236484

Offset: 1

Author

Janaka Rodrigo, May 22 2025

Keywords

• nonn

Comments

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.

Examples

			When n = 5,the duplet (5,5) can be decomposed in the following four different ways:
  {(1,1), (1,2), (1,4), (2,3), (3,4)},
  {(1,1), (1,3), (2,2), (2,4), (3,3)},
  {(1,2), (1,3), (1,4), (2,2), (3,4)},
  {(1,3), (1,4), (2,2), (2,3), (2,4)}.
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.
		

Crossrefs

Cf. A381847.