A387171 Number of 4 element sets of distinct integer sided rectangles that fill an n X n square.
0, 0, 0, 3, 15, 35, 75, 119, 210, 289, 441, 574, 804, 993, 1329, 1584, 2031, 2378, 2952, 3386, 4122, 4654, 5550, 6211, 7284, 8064, 9354, 10263, 11763, 12839, 14565, 15791, 17790, 19177, 21435, 23026, 25560, 27333, 30195, 32160, 35331, 37538, 41034, 43454, 47334
Offset: 1
Examples
The a(4) = 3 sets of integer sided rectangles are:
{(1 X 1), (3 X 1), (4 X 2), (4 X 1)},
{(2 X 1), (1 X 1), (3 X 3), (4 X 1)},
{(4 X 1), (2 X 3), (2 X 2), (2 X 1)}.
Links
- Janaka Rodrigo, Table of n, a(n) for n = 1..100
Formula
Conjectures from Vaclav Kotesovec, Aug 22 2025: (Start)
G.f.: x^4*(3 + 18*x + 47*x^2 + 86*x^3 + 105*x^4 + 107*x^5 + 77*x^6 + 45*x^7 + 17*x^8 + 5*x^9) / ((1-x)^4 * (1+x)^3 * (1+x^2) * (1+x+x^2)^2).
a(n) = -a(n-1) + a(n-2) + 3*a(n-3) + 3*a(n-4) - a(n-5) - 4*a(n-6) - 4*a(n-7) - a(n-8) + 3*a(n-9) + 3*a(n-10) + a(n-11) - a(n-12) - a(n-13).
a(6*n+3) = a(6*n-3) - 3*a(6*n-1) + 3*a(6*n+1) + 30.
For n > 0, a(n) = -5 + 1421*n/144 - 35*n^2/6 + 139*n^3/144 - floor(n/4)/4 + (-1 + 2*n/3)*floor(n/3) + (-27/8 + 29*n/8 - 3*n^2/4)*floor(n/2) - floor((1 + n)/4)/4 + (-2/3 + n/3)*floor((1 + n)/3).
a(n) ~ 85*n^3/144.
(End)