Janaka Rodrigo - cp's OEIS frontend

A384724 a(n) is the number of 4 element sets of distinct integer sided strict rectangles that fill an n X n square.

0, 0, 0, 0, 5, 15, 39, 70, 132, 197, 311, 421, 606, 771, 1047, 1275, 1655, 1968, 2466, 2863, 3510, 4004, 4802, 5416, 6384, 7116, 8286, 9144, 10517, 11535, 13125, 14290, 16140, 17465, 19565, 21085, 23454, 25155, 27837, 29727, 32711, 34836, 38136, 40471, 44142, 46700, 50720, 53548, 57936, 61008

Offset: 1

Janaka Rodrigo, Aug 23 2025

A strict rectangle is a rectangle that is not a square.

			a(5) = 5 sets of integer sided strict rectangles are:
  {(1,2), (1,3), (1,5), (3,5)},
  {(1,2), (1,3), (2,4), (3,4)},
  {(1,2), (1,5), (2,3), (3,4)},
  {(1,2), (1,5), (2,4), (2,5)},
  {(1,3), (1,4), (2,4), (2,5)}.

Cf. A387171.

A385240 Array read by descending antidiagonals: T(n,k) is the number of k element sets of noncongruent integer sided rectangles that fill an n X n square.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 1, 0, 0, 0, 3, 8, 2, 1, 0, 0, 0, 2, 15, 11, 3, 1, 0, 0, 0, 0, 19, 35, 19, 3, 1, 0, 0, 0, 0, 7, 87, 75, 23, 4, 1, 0, 0, 0, 0, 1, 114, 257, 119, 35, 4, 1, 0, 0, 0, 0, 0, 56, 593, 571, 210, 40, 5, 1, 0

Offset: 1

Janaka Rodrigo, Aug 26 2025

			Array begins:
  1     0     0     0     0
  1     0     0     0     0
  1     1     2     0     0
  1     1     3     3     2
  1     2     8    15    19
  1     2    11    35    87
  1     3    19    75   257
  1     3    23   119   571
  1     4    35   210  1186
  1     4    40   289  2033

Columns: A000012 (k=1), A004526 (k=2), A381847 (k=3), A387171 (k=4), A387241 (k=5).

Cf. A386296 (3-dimensional version).

T(n,1) = 1.

T(n,k) = 0 for k > n^2.

More terms from Sean A. Irvine, Sep 02 2025

A387121 Array read by antidiagonals: T(n,k) is the number of sets of k integer sided cuboids with distinct volumes that fill an n X n X n cube.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 4, 3, 2, 1, 0, 0, 2, 11, 8, 2, 1, 0, 0, 1, 26, 47, 11, 3, 1, 0, 0, 0, 55, 206, 77, 19, 3, 1, 0, 0, 0, 48, 793, 442, 183, 23, 4, 1, 0, 0, 0, 23, 2653, 2451, 1531, 259, 35, 4, 1, 0, 0, 0, 0, 6706, 13022, 12178

Offset: 1

Janaka Rodrigo, Aug 16 2025

The partitions here must be valid packings of the n X n X n cube, hence T(n,k) is generally less than the number of partitions of n^3 into distinct cuboids (x,y,z) with 1 <= x,y,z <= n and no pair of triplets having equal volume x*y*z.

			Array begins:
  1     0     0     0     0
  1     0     0     0     0
  1     1     2     4     2
  1     1     3    11    26
  1     2     8    47   206
  1     2    11    77   442
  1     3    19   183  1531
  1     3    23   259  2661
  1     4    35   457  5574
  1     4    40   599  8514
  ...

Columns are A004526 (k=2), A381847 (k=3), A385580 (k=4), A387040 (k=5).

T(n,1) = 1, T(n,k) = 0 for k > n^3.

More terms from Sean A. Irvine, Aug 25 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)

A386903 Array read by descending antidiagonals: T(n,k) is the number of ways to partition n X n X n cube into k noncongruent strict cuboids, n>=5, k>=4.

Original entry on oeis.org

1, 0, 2, 3, 18, 9, 1, 64, 74, 12, 1, 143, 450, 193, 30, 0, 197, 2090, 1769, 491, 36, 0, 156, 8039, 13441, 5687, 857, 70, 0, 57, 24641, 88001, 56540, 12994, 1695, 80, 0, 5

Offset: 5

Janaka Rodrigo, Aug 07 2025

A strict cuboid is a cuboid with all dimensions different.
The partitions here must be valid packings of the n X n X n cube, hence T(n,k) is generally less than the number of partitions of n^3 into distinct cuboids (x,y,z) with 1 <= x,y,z <= n, x != y != z and volume x*y*z.
There are no solutions for n < 5 or k < 4.

			Array begins:
  n\k|  4     5      6       7      8      9
  ---+--------------------------------------
   5 |  1     0      3       1      1      0
   6 |  2    18     64     143    197    156
   7 |  9    74    450    2090   8039  24641
   8 | 12   193   1769   13441  88001      ?
   9 | 30   491   5687   56540      ?      ?
  10 | 36   857  12994  170052      ?      ?
  ...

Cf. A386296.

Columns: A386884 (k=4), A386902 (k=5).

A387040 a(n) is the number of distinct five-cuboid combinations that fill an n X n X n cube with cuboids of different volumes.

Original entry on oeis.org

0, 0, 2, 26, 206, 442, 1531, 2661, 5574, 8514, 15614, 20331, 34500, 44814, 64503, 83143, 117759, 141290, 193436, 226722, 295978, 351953, 447208, 507508, 637447, 732322, 887044, 1001577, 1213233, 1337525, 1611692, 1786560, 2088648, 2321052, 2673275, 2929254, 3404667

Offset: 1

Janaka Rodrigo, Aug 14 2025

Alternatively a(n) is the number of ways to decompose (n,n,n) triplet into geometrically feasible five distinct unordered triplets of the form (x,y,z) with no pair having equal value for the product x*y*z.

			According to A384479(5), (5,5,5) triplet can be decomposed into 209 distinct sets of five triplets and only three of them contain pair of triplets with equal value for x*y*z. Those are,
   {(1,2,5), (1,3,5), (1,4,5), (2,2,5), (3,4,5)},
   {(1,1,5), (1,4,5), (2,2,5), (2,3,5), (2,5,5)},
   {(1,3,5), (1,4,5), (2,2,5), (2,3,5), (2,4,5)}.
Therefore a(5) = 209-3 = 206.

Cf. A384479, A385580, A387121.

a(15)-a(16) from Sean A. Irvine, Aug 19 2025

More terms from Jinyuan Wang, Aug 29 2025

A385580 a(n) is the number of ways to partition an n X n X n cube with four noncongruent cuboids of different volumes.

Original entry on oeis.org

0, 0, 4, 11, 47, 77, 183, 259, 457, 599, 941, 1120, 1668, 1986, 2637, 3125, 4079, 4622, 5868, 6530, 8061, 9028, 10874, 11856, 14148, 15522, 18074, 19583, 22739, 24292, 28065, 30105, 34071, 36544, 40885, 43520, 48888, 51912, 57512, 60666, 67331, 70777, 78078

Offset: 1

Janaka Rodrigo, Aug 12 2025

nonn

Alternatively, a(n) is the number of ways to decompose (n,n,n) triplet into geometrically feasible four distinct unordered triplets of the form (x,y,z) with no pairs of triplets having equal value for the product x*y*z.

			There are A384311(4) = 12 different ways to decompose a 4 X 4 X 4 cube into four noncongruent cuboids, but of those 12 ways, one partition {(4,2,1), (4,2,2), (4,3,2), (4,4,1)} contains two cuboids of volume 16 ((4,2,2) and (4,4,1)) which needs to be excluded. Therefore a(4) = 12-1 = 11.

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. A384311.

Conjecture: a(p) = A384311(p) for any prime p.

A386902 a(n) is the number of distinct five-cuboid combinations that fill an n X n X n with only strict cuboids.

Original entry on oeis.org

0, 0, 0, 0, 0, 18, 74, 193, 491, 857, 1695, 2503, 4321, 5836, 9200, 11715, 17284, 21256, 29805, 35589, 48156, 56260, 73766, 84860, 108495, 123080, 154298, 172998, 213045, 236895, 287260, 316743, 379465, 415456, 491930, 535713, 627879, 680052, 790401, 851914, 982130

Offset: 1

Janaka Rodrigo, Aug 07 2025

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

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

			(6,6,6) triplet can be decomposed into set of five triplets in 560 different ways and only 18 of those formed by only strict cuboids. Three of those sets are given below:
   {(1,2,3), (1,3,4), (2,3,6), (3,4,6), (3,5,6)},
   {(1,2,6), (1,4,6), (2,4,6), (2,5,6), (3,4,6)},
   {(1,3,4), (1,3,6), (2,3,5), (2,3,6), (4,5,6)}.

Cf. A384479, A386847.

a(16)-a(18) from Sean A. Irvine, Aug 14 2025

More terms from Jinyuan Wang, Aug 29 2025

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

Original entry on oeis.org

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

A386846 a(n) is the number of sets of distinct four-cuboid combinations that fill an n X n X n cube excluding combinations that contain strict cuboids.

Original entry on oeis.org

0, 0, 2, 3, 4, 6, 6, 9, 8, 13, 11, 17, 15, 23, 20, 30, 27, 39, 36, 50, 47, 64, 61, 80, 78, 100, 98, 123, 122, 150, 150, 181, 182, 217, 219, 257, 261, 303, 308, 354, 361, 411, 420, 474, 485, 544, 557, 620, 636, 704, 722, 795, 816, 894, 918, 1001, 1028, 1117

Offset: 1

Janaka Rodrigo, Aug 05 2025

nonn

A strict cuboid is a cuboid with all three dimensions different to each other.

Alternatively a(n) is number of ways to decompose (n,n,n) triplet into sets of distinct unordered geometrically feasible four triplets of the form (x,y,z) excluding x != y != z in any of the triplets.

			(5,5,5) triplet can be decomposed into sets of four triplets in 47 different ways and only the following 4 sets do not contain strict cuboids.
{(5,5,1), (5,4,4), (4,1,1), (1,4,4)},
{(5,5,3), (5,2,2), (3,3,2), (2,2,3)},
{(5,5,2), (3,3,5), (2,2,3), (3,3,2)},
{(4,1,1), (5,1,1), (1,4,4), (4,5,5)}.

Cf. A384311.

More terms from Sean A. Irvine, Aug 06 2025

cp's OEIS Frontend

User: Janaka Rodrigo

A384724 a(n) is the number of 4 element sets of distinct integer sided strict rectangles that fill an n X n square.

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A385240 Array read by descending antidiagonals: T(n,k) is the number of k element sets of noncongruent integer sided rectangles that fill an n X n square.

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A387121 Array read by antidiagonals: T(n,k) is the number of sets of k integer sided cuboids with distinct volumes that fill an n X n X n cube.

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

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A386903 Array read by descending antidiagonals: T(n,k) is the number of ways to partition n X n X n cube into k noncongruent strict cuboids, n>=5, k>=4.

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A387040 a(n) is the number of distinct five-cuboid combinations that fill an n X n X n cube with cuboids of different volumes.

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A385580 a(n) is the number of ways to partition an n X n X n cube with four noncongruent cuboids of different volumes.

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A386902 a(n) is the number of distinct five-cuboid combinations that fill an n X n X n with only strict cuboids.

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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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A386846 a(n) is the number of sets of distinct four-cuboid combinations that fill an n X n X n cube excluding combinations that contain strict cuboids.

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