A384311 -id:A384311 - cp's OEIS frontend

A386884 a(n) is the number of distinct four-cuboid combinations that fill an n X n X n cube using only strict cuboids.

0, 0, 0, 0, 1, 2, 9, 12, 30, 36, 70, 80, 135, 150, 231, 252, 364, 392, 540, 576, 765, 810, 1045, 1100, 1386, 1452, 1794, 1872, 2275, 2366, 2835, 2940, 3480, 3600, 4216, 4352, 5049, 5202, 5985, 6156, 7030, 7220, 8190, 8400, 9471, 9702, 10879, 11132, 12420

Offset: 1

Janaka Rodrigo, Aug 06 2025

nonn

A strict cuboid is a cuboid with all three dimensions different.

Alternatively a(n) is the number of ways to decompose triplet (n,n,n) into sets of distinct four unordered triplets of the form (x,y,z) with x != y != z for each of the four triplets.

			As described in A384311 there are 85 sets of distinct four-cuboid combinations filling 6 X 6 X 6 cube and only two of those have all four triplets with different elements, those are;
   {(1,2,6), (1,4,6), (2,5,6), (4,5,6)},
   {(1,3,6), (2,3,6), (3,4,6), (3,5,6)}.
Therefore a(6) = 2.

Cf. A384311.

More terms from Sean A. Irvine, Aug 06 2025

A387171 Number of 4 element sets of distinct integer sided rectangles that fill an n X n square.

Original entry on oeis.org

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

Janaka Rodrigo, Aug 20 2025

			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)}.

Janaka Rodrigo, Table of n, a(n) for n = 1..100

Column 4 of A385240.

Cf. A384311 (3-dimensional version).

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)

cp's OEIS Frontend

A386884 a(n) is the number of distinct four-cuboid combinations that fill an n X n X n cube using only strict cuboids.

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