cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A034295 Number of different ways to divide an n X n square into sub-squares, considering only the list of parts.

Original entry on oeis.org

1, 2, 3, 7, 11, 31, 57, 148, 312, 754, 1559, 3844, 7893, 17766, 37935, 83667, 170165, 369698, 743543, 1566258, 3154006, 6424822, 12629174, 25652807, 49802454, 98130924, 189175310, 368095797
Offset: 1

Views

Author

Erich Friedman, Dec 11 1999

Keywords

Comments

Number of ways an n X n square can be cut into integer-sided squares: collections of integers {a_i} so that squares of length a_i tile an n X n square.
This ignores the way the squares are arranged. We are only counting the lists of parts (compare A045846).
Also applies to the partitions of an equilateral triangle of length n. - Robert G. Wilson v

Examples

			From _Jon E. Schoenfield_, Sep 18 2008: (Start)
a(3) = 3 because the 3 X 3 square can be divided into sub-squares in 3 different ways: a single 3 X 3 square, a 2 X 2 square plus five 1 X 1 squares, or nine 1 X 1 squares.
There are a(5) = 11 different ways to divide a 5 X 5 square into sub-squares:
   1. 25(1 X 1)
   2.  1(2 X 2) + 21(1 X 1)
   3.  2(2 X 2) + 17(1 X 1)
   4.  3(2 X 2) + 13(1 X 1)
   5.  4(2 X 2) +  9(1 X 1)
   6.  1(3 X 3) + 16(1 X 1)
   7.  1(3 X 3) +  1(2 X 2) + 12(1 X 1)
   8.  1(3 X 3) +  2(2 X 2) +  8(1 X 1)
   9.  1(3 X 3) +  3(2 X 2) +  4(1 X 1)
  10.  1(4 X 4) +  9(1 X 1)
  11.  1(5 X 5)
a(9) = 312 because the 9 X 9 square can be divided into 312 different combinations of sub-squares such as three 4 X 4 squares plus thirty-three 1 X 1 squares, etc. (End)

Links

Crossrefs

Cf. A045846, A224239.
Cf. A014544, A129668 (these both involve cubes).
Main diagonal of A224697.

Programs

  • Maple
    b:= proc(n, l) option remember; local i, k, s;
      if max(l[])>n then {} elif n=0 then {0}
    elif min(l[])>0 then (t->b(n-t, map(h->h-t, l)))(min(l[]))
    else for k while l[k]>0 do od; s:={};
         for i from k to nops(l) while l[i]=0 do s:=s union
             map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
                 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
a:= n-> nops(b(n, [0$n])):
seq(a(n), n=1..9);  # Alois P. Heinz, Apr 15 2013
  • Mathematica
    $RecursionLimit = 1000; b[n_, l_] := b[n, l] = Module[{i, k, m, s, t}, Which[Max[l]>n, {}, n == 0 || l == {}, {{}}, Min[l]>0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[x, Sort[Append[x, 1+i-k]]], b[n, Join[l[[1 ;; k-1]], Array[1+i-k &, i-k+1], l[[i+1 ;; -1]]]]]]; s]]; a[n_] := a[n] = b[n, Array[0&, n]] // Length; Table[Print[a[n]]; a[n], {n, 1, 12} ] (* Jean-François Alcover, Feb 18 2014, after Alois P. Heinz *)

Extensions

More terms from Sergio Pimentel, Jun 03 2008
Corrected and extended by Jon E. Schoenfield, Sep 19 2008
Edited by N. J. A. Sloane, Apr 12 2013, at the suggestion of Paolo P. Lava
a(11) corrected by Alois P. Heinz, Apr 15 2013
a(13) from Alois P. Heinz, Apr 19 2013
a(14) from Christopher Hunt Gribble, Oct 26 2013
a(15) and a(16) from Fidel I. Schaposnik, May 04 2015
a(17)-a(23) from Holger Langenau, Sep 20 2017
a(24) from Michael De Vlieger, May 04 2018, from paper written by Holger Langenau
a(25) and a(26) from Holger Langenau, May 14 2018
a(27) from Holger Langenau, Apr 15 2019
a(28) from Holger Langenau, Jun 17 2020
a(28) corrected by Holger Langenau, Jul 31 2020

A227690 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles reduced for symmetry; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 5, 6, 6, 5, 1, 1, 1, 1, 9, 10, 13, 10, 9, 1, 1, 1, 1, 12, 21, 39, 39, 21, 12, 1, 1, 1, 1, 21, 39, 115, 77, 115, 39, 21, 1, 1, 1, 1, 30, 82, 295, 521, 521, 295, 82, 30, 1, 1
Offset: 0

Views

Author

Christopher Hunt Gribble, Jul 19 2013

Keywords

Examples

			Square array A(n,k) begins:
  1, 1,  1,  1,   1,    1,     1,      1,       1, ...
  1, 1,  1,  1,   1,    1,     1,      1,       1, ...
  1, 1,  2,  2,   4,    5,     9,     12,      21, ...
  1, 1,  2,  3,   6,   10,    21,     39,      82, ...
  1, 1,  4,  6,  13,   39,   115,    295,     861, ...
  1, 1,  5, 10,  39,   77,   521,   1985,    8038, ...
  1, 1,  9, 21, 115,  521,  1494,  15129,   83609, ...
  1, 1, 12, 39, 295, 1985, 15129,  56978,  861159, ...
  1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023, ...
...
A(4,3) = 6 because there are 6 ways to tile a 3 X 4 rectangle by subsquares, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct:
   ._____ _.    ._______.    ._______.
   |     |_|    |   |   |    |   |_|_|
   |     |_|    |___|_ _|    |___|   |
   |_____|_|    |_|_|_|_|    |_|_|___|
   ._______.    ._______.    ._______.
   |   |_|_|    |_|   |_|    |_|_|_|_|
   |___|_|_|    |_|___|_|    |_|_|_|_|
   |_|_|_|_|    |_|_|_|_|    |_|_|_|_|

Links

Crossrefs

Main diagonal: A224239.
Columns 1-10: A000012, A001224, A359019, A359020, A359021, A359022, A359023, A359024, A359025, A359026.
Cf. A219924, A224697.

A360629 Triangle read by rows: T(n,k) is the number of sets of integer-sided rectangular pieces that can tile an n X k rectangle, 1 <= k <= n.

Original entry on oeis.org

1, 2, 4, 3, 10, 21, 5, 22, 73, 192, 7, 44, 190, 703, 2035, 11, 91, 510, 2287, 8581, 27407, 15, 172, 1196, 6738, 30209, 118461, 399618, 22, 326, 2895, 19160, 102092, 462114
Offset: 1

Views

Author

Pontus von Brömssen, Feb 14 2023

Keywords

Comments

Pieces are free to rotate by 90 degrees, i.e., an r X s piece and an s X r piece are equivalent. See A360451 for the case when the pieces are fixed.

Examples

			Triangle begins:
   n\k|  1   2    3    4     5      6      7
   ---+--------------------------------------
   1  |  1
   2  |  2   4
   3  |  3  10   21
   4  |  5  22   73  192
   5  |  7  44  190  703  2035
   6  | 11  91  510 2287  8581  27407
   7  | 15 172 1196 6738 30209 118461 399618
   ...
T(2,2) = 4, because a 2 X 2 rectangle can be tiled by: one 2 X 2 piece; two 1 X 2 pieces; one 1 X 2 piece and two 1 X 1 pieces; four 1 X 1 pieces.
The T(3,2) = 10 sets of pieces that can tile a 3 X 2 rectangle are shown in the table below. (Each column on the right gives a set of pieces.)
   length X width |  number of pieces
   ---------------+--------------------
        2 X 3     | 1 0 0 0 0 0 0 0 0 0
        2 X 2     | 0 1 1 0 0 0 0 0 0 0
        1 X 3     | 0 0 0 2 1 1 0 0 0 0
        1 X 2     | 0 1 0 0 1 0 3 2 1 0
        1 X 1     | 0 0 2 0 1 3 0 2 4 6

Crossrefs

Cf. A000041 (column k=1), A116694, A224697 (square pieces), A360451 (fixed pieces), A360630 (main diagonal), A360631 (column k=2), A360632 (column k=3).

Extensions

T(7,7) and T(8,k) for k = 1..6 added by Robin Visser, May 09 2025

A225542 Number T(n,k,u) of partitions of an n X k rectangle into integer-sided square parts containing u nodes that are unconnected to any of their neighbors, considering only the number of parts; irregular triangle T(n,k,u), 1 <= k <= n, u >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 28 2013

Keywords

Comments

The number of entries per row is given by A225568.

Examples

			The irregular triangle begins:
n,k\u 0   1   2   3   4   5   6   7   8   9  10  11  12 ...
1,1   1
2,1   1
2,2   1   1
3,1   1
3,2   1   1
3,3   1   1   0   0   1
4,1   1
4,2   1   1   1
4,3   1   1   1   0   1
4,4   1   1   1   1   2   0   0   0   0   1
5,1   1
5,2   1   1   1
5,3   1   1   1   0   1   1
5,4   1   1   1   1   2   1   1   0   0   1
5,5   1   1   1   1   2   1   1   1   0   1   0   0   0 ...
...
For n = 5 and k = 4 there are 2 partitions that contain 4 isolated nodes, so T(5,4,4) = 2.
Consider that each partition is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  Then the 2 partitions are:
1 1 1 1 1    1 1 1 1 1
1 0 1 0 1    1 0 0 1 1
1 1 1 1 1    1 0 0 1 1
1 0 1 0 1    1 1 1 1 1
1 1 1 1 1    1 1 1 1 1
1 1 1 1 1    1 1 1 1 1

Links

Crossrefs

Cf. A034295, A224697, A227009, A225777, A225803, A225568.

Formula

T(n,n,u) = A227009(n,u).
Sum_{u=1..(n-1)^2} T(n,n,u) = A034295(n).

A362142 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 3, 6, 16, 1, 1, 4, 12, 37, 140, 1, 1, 6, 24, 105, 454, 1987, 1, 1, 10, 40, 250, 1566, 9856, 62266, 1, 1, 15, 80, 726, 5670, 47394, 406168, 3899340, 1, 1, 21, 160, 1824, 18738, 223696, 2916492, 38322758, 508317004
Offset: 0

Views

Author

Pontus von Brömssen, Apr 10 2023

Keywords

Examples

			Triangle begins:
  n\k| 0  1  2  3   4    5     6      7       8
  ---+-----------------------------------------
  0  | 1
  1  | 1  1
  2  | 1  1  1
  3  | 1  1  2  4
  4  | 1  1  3  6  16
  5  | 1  1  4 12  37  140
  6  | 1  1  6 24 105  454  1987
  7  | 1  1 10 40 250 1566  9856  62266
  8  | 1  1 15 80 726 5670 47394 406168 3899340
A 5 X 4 rectangle can be tiled by 12 unit squares and 2 squares of side 2 in the following ways:
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+---+---+   +       +---+---+   +---+       +---+   +---+---+       +
  |   |   |       |   |       |   |   |   |   |       |   |   |   |   |       |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |       |   |       |   |   |   |       |   |   |   |       |   |   |
  +       +---+---+   +       +---+---+   +       +---+---+   +       +---+---+
  |       |   |   |   |       |   |   |   |       |   |   |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |       |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |
  +---+       +---+   +---+---+---+---+   +---+---+---+---+   +       +---+---+
  |   |       |   |   |       |   |   |   |   |   |   |   |   |       |   |   |
  +---+---+---+---+   +       +---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+       +   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |   |       |   |   |       |   |   |   |   |   |   |   |   |   |       |
  +---+---+---+---+   +---+       +---+   +---+---+---+---+   +---+---+       +
  |   |   |   |   |   |   |       |   |   |       |       |   |       |       |
  +---+---+---+---+   +---+---+---+---+   +       +       +   +       +---+---+
  |       |   |   |   |   |       |   |   |       |       |   |       |   |   |
  +       +---+---+   +---+       +---+   +---+---+---+---+   +---+---+---+---+
  |       |   |   |   |   |       |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
  |   |   |   |   |
  +---+---+---+---+
  |   |       |   |
  +---+       +---+
  |   |       |   |
  +---+---+---+---+
The first six of these have no symmetries, so they account for 4 tilings each. The next six have either a mirror symmetry or a rotational symmetry and account for 2 tilings each. The last has full symmetry and accounts for 1 tiling. In total there are 6*4+6*2+1 = 37 tilings. This is the maximum for a 5 X 4 rectangle, so T(5,4) = 37.

Links

Crossrefs

Main diagonal: A362143.
Columns: A000012 (k = 0,1), A073028 (k = 2), A362144 (k = 3), A362145 (k = 4), A362146 (k = 5).
Cf. A219924, A224697, A361216 (rectangular pieces).

A224850 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 1 element; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 5, 2, 12, 6, 1, 1, 3, 3, 5, 7, 17, 1, 1, 8, 3, 25, 11, 106, 44
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 22 2013

Keywords

Comments

It appears that sequence T(2,k) consists of 2 interspersed Fibonacci sequences.
The diagonal T(n,n) is A006081. - M. F. Hasler, Jul 25 2013

Examples

			The triangle is:
n\k  1   2   3   4   5   6   7   8 ...
.
0    1   1   1   1   1   1   1   1 ...
1        1   1   1   1   1   1   1 ...
2            1   3   2   5   3   8 ...
3                1   2   2   3   3 ...
4                    3  12   5  25 ...
5                        6   7  11 ...
6                           17 106 ...
7                               44 ...
...
T(3,5) = 2 because there are 2 different tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will only transform each tiling into itself.  Group D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
The tilings are:
._________.    ._________.
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|_____|_|

Links

Crossrefs

Cf. A219924, A224697, A227690.

Formula

T(n,k) + A224861(n,k) + A224867(n,k) = A227690(n,k).
1*T(n,k) + 2*A224861(n,k) + 4*A224867(n,k) = A219924(n,k).

A224861 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 2 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 3, 3, 15, 0, 0, 4, 9, 38, 75, 0, 0, 9, 9, 68, 77, 604, 0, 0, 13, 21, 160, 311, 2384, 4556
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 22 2013

Keywords

Examples

			The triangle is:
n\k  1   2   3   4   5   6   7    8 ...
.
0    0   0   0   0   0   0   0    0 ...
1        0   0   0   0   0   0    0 ...
2            1   1   3   4   9   13 ...
3                4   3   9   9   21 ...
4                   15  38  68  160 ...
5                       75  77  311 ...
6                          604 2384 ...
7                              4556 ...
...
T(3,5) = 3 because there are 3 different sets of 2 tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will transform each tiling in a set into the other in the same set.  Group D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
An example of a tiling in each set is:
._________.    ._________.    ._________.
|   |_|   |    |   |_|_|_|    |     |_|_|
|_ _|_|_ _|    |___|_|   |    |     |_|_|
|_|_|_|_|_|    |_|_|_|___|    |_____|_|_|

Links

Crossrefs

Cf. A219924, A224697, A227690.

Formula

A224850(n,k) + T(n,k) + A224867(n,k) = A227690(n,k).
1*A224850(n,k) + 2*T(n,k) + 4*A224867(n,k) = A219924(n,k).

A224867 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 4 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 21, 0, 0, 0, 10, 65, 440, 0, 0, 0, 27, 222, 1901, 14508, 0, 0, 0, 58, 676, 7716, 81119, 856559
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 22 2013

Keywords

Examples

			The triangle is:
n\k    1      2      3      4      5      6      7      8 ...
.
0      0      0      0      0      0      0      0      0 ...
1             0      0      0      0      0      0      0 ...
2                    0      0      0      0      0      0 ...
3                           1      5     10     27     58 ...
4                                 21     65    222    676 ...
5                                       440   1901   7716 ...
6                                            14508  81119 ...
7                                                  856559 ...
...
T(3,5) = 5 because there are 5 different sets of 4 tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will transform each tiling in a set into another in the same set.  Group  D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
An example of a tiling in each set is:
._________.  ._________.  ._________.  ._________.  ._________.
|   |_|_|_|  |_|   |_|_|  |   |   |_|  |   |_|_|_|  |   |     |
|_ _|_|_|_|  |_|_ _|_|_|  |_ _|_ _|_|  |___|   |_|  |___|     |
|_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|___|_|  |_|_|_____|

Links

Crossrefs

Cf. A219924, A224697, A227690.

Formula

A224850(n,k) + A224861(n,k) + T(n,k) = A227690(n,k).
1*A224850(n,k) + 2*A224861(n,k) + 4*T(n,k) = A219924(n,k).

A225622 A(n,k) is the total number of parts in the set of partitions of an n X k rectangle into integer-sided squares, considering only the list of parts; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 5, 15, 16, 15, 5, 6, 21, 31, 31, 21, 6, 7, 30, 47, 59, 47, 30, 7, 8, 38, 73, 102, 102, 73, 38, 8, 9, 50, 101, 170, 156, 170, 101, 50, 9, 10, 60, 142, 250, 307, 307, 250, 142, 60, 10, 11, 75, 185, 375, 460, 529, 460, 375, 185, 75, 11
Offset: 1

Views

Author

Christopher Hunt Gribble, Aug 04 2013

Keywords

Examples

			The square array starts:
1    2    3    4    5    6    7    8    9   10   11   12 ...
2    5    9   15   21   30   38   50   60   75   87  105 ...
3    9   16   31   47   73  101  142  185  244  305  386 ...
4   15   31   59  102  170  250  375  523  726  962 ...
5   21   47  102  156  307  460  711 1040 1517 ...
6   30   73  170  307  529  907 1474 2204 ...
7   38  101  250  460  907 1351 2484 ...
8   50  142  375  711 1474 2484 ...
9   60  185  523 1040 2204 ...
...
A(3,2) = 9 because there are 9 parts overall in the 2 partitions of a 3 X 2 rectangle into squares with integer sides.  One partition comprises 6 1 X 1 squares and the other 2 1 X 1 squares and 1 2 X 2 square giving 9 parts in total.

Links

Crossrefs

Diagonal = A226897.
Cf. A034295, A224697, A225542.

Programs

  • Maple
    b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then {} elif n=0 or l=[] then {0}
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:={};
         for i from k to nops(l) while l[i]=0 do s:=s union
             map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
                 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> add(coeff(add(j, j=b(max(n, k),
            [0$min(n, k)])), x, i), i=1..n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..15); # Alois P. Heinz, Aug 04 2013
  • Mathematica
    $RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[ Max[l]>n, {}, n == 0 || l == {}, {0}, Min[l]>0, t = Min[l]; b[n-t, l-t], True, For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[{v}, v + x^(1+i-k) ], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]]]]]]; s]];A[n_, k_] := Sum[Coefficient[Sum[j, {j, b[Max[n, k], Array[0&, Min[n, k]]]}], x, i], {i, 1, n}]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 15}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

Formula

A(n,1) = A000027(n) = n.
A(n,2) = A195014(n) = (n+1)(5n+3)/8 when n is odd
and 5n(n+2)/8 when n is even.

A228202 Number T(n,k,r) of partitions of an n X k X r rectangular cuboid into integer-sided cubes, considering only the list of parts; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 3, 1, 3, 4, 1, 5, 6, 11, 1, 1, 3, 1, 3, 5, 1, 5, 8, 15, 1, 5, 9, 17, 19, 1, 1, 4, 1, 4, 7, 1, 7, 11, 24, 1, 7, 16, 34, 40, 1, 10, 23, 52, 80, 121
Offset: 1

Views

Author

Christopher Hunt Gribble, Aug 17 2013

Keywords

Examples

			The irregular triangle begins:
   r  1   2   3   4 ...
n,k
1,1   1
2,1   1
2,2   1   2
3,1   1
3,2   1   2
3,3   1   2   3
4,1   1
4,2   1   3
4,3   1   3   4
4,4   1   5   6  11
5,1   1
5,2   1   3
5,3   1   3   5
5,4   1   5   8  15
5,5   1   5   9  17  19
...
T(4,4,3) = 6 because there are 6 partitions of a 4 X 4 X 3 rectangular cuboid into integer-sided cubes.  The partitions are:
48 1 X 1 X 1 cubes,
40 1 X 1 X 1 cubes and 1 2 X 2 X 2 cube,
32 1 X 1 X 1 cubes and 2 2 X 2 X 2 cubes,
24 1 X 1 X 1 cubes and 3 2 X 2 X 2 cubes,
16 1 X 1 X 1 cubes and 4 2 X 2 X 2 cubes,
21 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube.

Links

Crossrefs

T(n,n,n) = A129668(n).
Cf. A224697.

Extensions

21 more terms (row 6) from Alois P. Heinz, Aug 18 2013
Showing 1-10 of 16 results. Next

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