A045846 -id:A045846 - cp's OEIS frontend

A063443 Number of ways to tile an n X n square with 1 X 1 and 2 X 2 tiles.

1, 1, 2, 5, 35, 314, 6427, 202841, 12727570, 1355115601, 269718819131, 94707789944544, 60711713670028729, 69645620389200894313, 144633664064386054815370, 540156683236043677756331721, 3641548665525780178990584908643, 44222017282082621251230960522832336

Offset: 0

Reiner Martin, Jul 23 2001

a(n) is also the number of ways to populate an n-1 X n-1 chessboard with nonattacking kings (including the case of zero kings). Cf. A193580. - Andrew Woods, Aug 27 2011

Also the number of vertex covers and independent vertex sets of the n-1 X n-1 king graph.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 343

Andrew Woods and Vaclav Kotesovec and Johan Nilsson, Table of n, a(n) for n = 0..40 (terms 0..21 from Andrew Woods, terms 22..24 from Vaclav Kotesovec and terms 25..40 from Johan Nilsson)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 68-69.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO], 2016, Section 4.1.
J. Nilsson, On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.2.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A001045, A006506, A054854, A054855, A063650-A063653, A067966, etc.
Cf. A045846, A211348, A247413, A201513.
Cf. A212269, A067958.
a(n) = row sum n-1 of A193580.
Main diagonal of A245013.

Needs["LinearAlgebra`MatrixManipulation`"] Remove[mat] step[sa[rules1_, {dim1_, dim1_}], sa[rules2_, {dim2_, dim2_}]] := sa[Join[rules2, rules1 /. {x_Integer, y_Integer} -> {x + dim2, y}, rules1 /. {x_Integer, y_Integer} -> {x, y + dim2}], {dim1 + dim2, dim1 + dim2}] mat[0] = sa[{{1, 1} -> 1}, {1, 1}]; mat[1] = sa[{{1, 1} -> 1, {1, 2} -> 1, {2, 1} -> 1}, {2, 2}]; mat[n_] := mat[n] = step[mat[n - 2], mat[n - 1]]; A[n_] := mat[n] /. sa -> SparseArray; F[n_] := MatrixPower[A[n], n + 1][[1, 1]]; (* Mark McClure (mcmcclur(AT)bulldog.unca.edu), Mar 19 2006 *)
$RecursionLimit = 1000; Clear[a, b]; b[n_, l_List] := b[n, l] = Module[{m=Min[l], k}, If[m>0, b[n-m, l-m], If[n == 0, 1, k=Position[l, 0, 1, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 2}]], 0]]]]; a[n_] := a[n] = If[n<2, 1, b[n, Table[0, {n}]]]; Table[Print[a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 11 2014, after Alois P. Heinz *)

Lim_{n -> infinity} (a(n))^(1/n^2) = A247413 = 1.342643951124... . - Brendan McKay, 1996

4 more terms from R. H. Hardin, Jan 23 2002
2 more terms from Keith Schneider (kschneid(AT)bulldog.unca.edu), Mar 19 2006
5 more terms from Andrew Woods, Aug 27 2011
a(22)-a(24) in b-file from Vaclav Kotesovec, May 01 2012
a(0) inserted by Alois P. Heinz, Sep 17 2014
a(25)-a(40) in b-file from Johan Nilsson, Mar 10 2016

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

Erich Friedman, Dec 11 1999

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

			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)

Alois P. Heinz, List of different ways to divide a 13 X 13 square into sub-squares
Alois P. Heinz, More ways to divide an 11 X 11 square into sub-squares
Holger Langenau, Squaring the square: New methods for determining the number of perfect square packings, 2018.
Jon E. Schoenfield, Table of solutions for n <= 12 (Caution: this table is missing 6 of the ways to divide an 11 X 11 square into sub-squares! Please see the Alois P. Heinz link listing those six ways. Thanks to Alois for catching this! -- Jon E. Schoenfield)
N. J. A. Sloane, Drawing to illustrate a(1)-a(4)

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

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

$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 *)

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

A219924 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles; 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, 3, 3, 1, 1, 1, 1, 5, 6, 5, 1, 1, 1, 1, 8, 13, 13, 8, 1, 1, 1, 1, 13, 28, 40, 28, 13, 1, 1, 1, 1, 21, 60, 117, 117, 60, 21, 1, 1, 1, 1, 34, 129, 348, 472, 348, 129, 34, 1, 1, 1, 1, 55, 277, 1029, 1916, 1916, 1029, 277, 55, 1, 1

Offset: 0

Alois P. Heinz, Dec 01 2012

For drawings of A(1,1), A(2,2), ..., A(5,5) see A224239.

			A(3,3) = 6, because there are 6 tilings of a 3 X 3 rectangle using integer-sided squares:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |     |  |   |_|  |_|   |  |_|_|_|  |_|_|_|  |_|_|_|
  |     |  |___|_|  |_|___|  |_|   |  |   |_|  |_|_|_|
  |_____|  |_|_|_|  |_|_|_|  |_|___|  |___|_|  |_|_|_|
Square array A(n,k) begins:
  1,  1,  1,   1,    1,    1,     1,      1, ...
  1,  1,  1,   1,    1,    1,     1,      1, ...
  1,  1,  2,   3,    5,    8,    13,     21, ...
  1,  1,  3,   6,   13,   28,    60,    129, ...
  1,  1,  5,  13,   40,  117,   348,   1029, ...
  1,  1,  8,  28,  117,  472,  1916,   7765, ...
  1,  1, 13,  60,  348, 1916, 10668,  59257, ...
  1,  1, 21, 129, 1029, 7765, 59257, 450924, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened
Steve Butler, Jason Ekstrand, Steven Osborne, Counting Tilings by Taking Walks in a Graph, A Project-Based Guide to Undergraduate Research in Mathematics, Birkhäuser, Cham (2020), see page 169.

Columns (or rows) k=0+1, 2-10 give: A000012, A000045(n+1), A002478, A054856, A054857, A219925, A219926, A219927, A219928, A219929.
Main diagonal gives A045846.
Cf. A113881, A226545.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    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:=0;
         for i from k to nops(l) while l[i]=0 do s:=s+
           b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);
# The following is a second version of the program that lists the actual dissections. It produces a list of pairs for each dissection:
b:= proc(n, l, ll) local i, k, s, t;
      if max(l[])>n then 0 elif n=0 or l=[] then lprint(ll); 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l), ll)
    else for k do if l[k]=0 then break fi od; s:=0;
         for i from k to nops(l) while l[i]=0 do s:=s+
           b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)],
            [ll[],[k,1+i-k]])
         od; s
      fi
    end:
A:= (n, k)-> b(k, [0$n], []):
A(5,5);
# In each list [a,b] means put a square with side length b at
leftmost possible position with upper corner in row a.  For example
[[1,3], [4,2], [4,2], [1,2], [3,1], [3,1], [4,1], [5,1]], gives:
 ___.___.
|     |   |
|     |_|
|___|_|_|
|   |   |_|
|_|___|_|

b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = 0; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join[l[[1;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1;; -1]] ] ] ]; s]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from 1st Maple program *)

A224239 Number of inequivalent ways to cut an n X n square into squares with integer sides.

Original entry on oeis.org

1, 2, 3, 13, 77, 1494, 56978, 4495023, 669203528, 187623057932, 98793520541768, 97702673827558670

Offset: 1

N. J. A. Sloane, Apr 15 2013

Similar to A045846, but now we do not regard dissections which differ by a rotation and/or reflection as distinct.

			For n=5, the illustrations (see links) show that the 77 solutions consist of:
4 dissections each with 1 image under the group of the square, for a total of 4,
2 dissections each with 2 images under the group of the square, totaling 4,
26 dissections each with 4 images under the group of the square, totaling 104, and
45 dissections each with 8 images under the group of the square, totaling 360,
for a grand total of 77 dissections with 472 images, agreeing with A045846(5) = 472.

Don Reble, C programs for A224239
Don Reble, Comments on the calculation of a(10)
N. J. A. Sloane, Illustration of the first five terms, page 1 of 4 (Each dissection is labeled with the number of its images under the symmetry group of the square. The sum of these numbers is A045846(n).)
N. J. A. Sloane, Illustration of the first five terms, page 2 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
N. J. A. Sloane, Illustration of the first five terms, page 3 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
N. J. A. Sloane, Illustration of the first five terms, page 4 of 4 (The largest squares are drawn in red. The next-largest squares, unless of size 1, are drawn in blue.)
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, 2013; arXiv:1308.5420

Main diagonal of A227690.

Cf. A045846, A034295, A219924.

a(6)-a(10) from Don Reble, Apr 15 2013

a(11)-a(12) from Ed Wynn, 2013. - N. J. A. Sloane, Nov 29 2013

A226979 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 2 elements.

Original entry on oeis.org

0, 0, 0, 2, 2, 24, 36, 344, 504, 7657, 11978, 289829

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

			For n=5, there are 2 dissections where the orbits under the symmetry group of the square, D4, have 2 elements.
For n=4, the 2 dissections can be seen in A240120 and A240121.

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A045846, A034295, A219924, A224239, A226978, A226980, A226981, A240120, A240121, A240122.

A226978(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).
1*A226978(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).
A226979(n) = A240120(n) + A240121(n) + A240122(n).

a(8)-a(12) from Ed Wynn, Apr 01 2014

A226980 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

Original entry on oeis.org

0, 0, 1, 6, 26, 264, 1157, 23460, 153485, 6748424, 70521609, 6791578258

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

			For n=5, there are 26 dissections where the orbits under the symmetry group of the square, D4, have 4 elements.
The 6 dissections for n=4 can be seen in A240123 and A240125.

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A045846, A034295, A219924, A224239, A226978, A226979, A226981, A240123, A240124, A240125.

A226978(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).
1*A226978(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).
A226980(n) = A240123(n) + A240124(n) + A240125(n).

a(8)-a(12) from Ed Wynn, Apr 01 2014

A225777 Number T(n,k,u) of distinct tilings of an n X k rectangle using integer-sided square tiles containing u nodes that are unconnected to any of their neighbors; 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, 2, 1, 4, 0, 0, 1, 1, 1, 3, 1, 1, 6, 4, 0, 2, 1, 9, 16, 8, 5, 0, 0, 0, 0, 1, 1, 1, 4, 3, 1, 8, 12, 0, 3, 4, 1, 12, 37, 34, 15, 12, 4, 0, 0, 2, 1, 16, 78, 140, 88, 44, 68, 32, 0, 4, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Christopher Hunt Gribble, Jul 26 2013

The number of entries per row is given by A225568.

			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   2
3,3   1   4   0   0   1
4,1   1
4,2   1   3   1
4,3   1   6   4   0   2
4,4   1   9  16   8   5   0   0   0   0   1
5,1   1
5,2   1   4   3
5,3   1   8  12   0   3   4
5,4   1  12  37  34  15  12   4   0   0   2
5,5   1  16  78 140  88  44  68  32   0   4   0   0   0 ...
...
For n = 4, k = 3, there are 4 tilings that contain 2 isolated nodes, so T(4,3,2) = 4. A 2 X 2 square contains 1 isolated node.  Consider that each tiling 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 4 tilings are:
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 0 1 1    1 1 0 1    1 1 0 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 1 0 1    1 0 1 1    1 1 0 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1

Christopher Hunt Gribble, Rows 1..36 for n=2..8 and k=1..n flattened
Christopher Hunt Gribble, C++ program

Cf. A045846, A219924, A226997, A225542, A225803, A225568.

T(n,k,0) = 1, T(n,k,1) = (n-1)(k-1), T(n,k,2) = (n^2(k-1) - n(2k^2+5k-13) + (k^2+13k-24))/2.

Sum_{u=1..(n-1)^2} T(n,n,u) = A045846(n).

A226978 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element.

Original entry on oeis.org

1, 2, 2, 4, 4, 12, 8, 44, 32, 228, 148, 1632, 912, 16004, 8420, 213680, 101508, 3933380, 1691008, 98949060, 38742844, 3413919788, 1213540776, 161410887252, 52106993880

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

From Walter Trump, Dec 15 2022: (Start)
a(n) is the number of fully symmetric dissections of an n X n square into squares with integer sides.
Conjecture: For n>3 the number of dissections is a multiple of 4. (End)

			For n=5, there are 4 dissections where the orbits under the symmetry group of the square, D4, have 1 element.
For n=4, 3 dissections divide the square into uniform subsquares (of sizes 1, 2 and 4 respectively), and this is the 4th:
---------
| | | | |
---------
| |   | |
---   ---
| |   | |
---------
| | | | |
---------

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Walter Trump, Example for n=19

Cf. A045846, A034295, A219924, A224239, A226979, A226980, A226981.

a(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).

1*a(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).

a(8)-a(12) from Ed Wynn, Apr 02 2014

a(13)-a(25) from Walter Trump, Dec 15 2022

A226981 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 8 elements.

Original entry on oeis.org

0, 0, 0, 1, 45, 1194, 55777, 4471175, 669049507, 187616301623, 98793450008033, 97702667035688951

Offset: 1

Christopher Hunt Gribble, Jun 25 2013

			For n=5, there are 45 dissections where the orbits under the symmetry group of the square, D4, have 8 elements.
For n=4, this is the only dissection:
---------
|   | | |
|   -----
|   |   |
-----   |
| | |   |
---------
| | | | |
---------

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420

Cf. A045846, A034295, A219924, A224239, A226978, A226979, A226980.

A226978(n) + A226979(n) + A226980(n) + A226981(n) = A224239(n).

1*A226978(n) + 2*A226979(n) + 4*A226980(n) + 8*A226981(n) = A045846(n).

a(8)-a(12) from Ed Wynn, Apr 02 2014

A226554 Number of squares in all tilings of an n X n square using integer-sided square tiles.

Original entry on oeis.org

0, 1, 5, 34, 386, 6940, 221672, 12582472, 1293374998, 242394178200, 83374069529638, 52845726291860344, 61928161880183204434, 134499571879749571406816, 542432658409586214809714176, 4068438590479352629770422328000, 56820656114941381799512710314429768

Offset: 0

Alois P. Heinz, Jun 10 2013

nonn

Main diagonal of A226545.
Row sums of A226936.
Cf. A045846.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then [0, 0] elif n=0 or l=[] then [1, 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:=[0$2];
         for i from k to nops(l) while l[i]=0 do s:=s+(h->h+[0, h[1]])
           (b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
a:= n-> b(n, [0$n])[2]:
seq(a(n), n=0..10);

b[n_, l_] := b[n, l] = Module[{i, k, s, t},
     Which[Max[l] > n, {0, 0}, n == 0 || l == {}, {1, 0},
     Min[l] > 0, t = Min[l]; b[n - t, l - t], True,
       k = Position[l, 0, 1][[1, 1]]; s = {0, 0};
     For[i = k, i <= Length[l] && l[[i]] == 0, i++,
       s = s + Function[h, h + {0, h[[1]]}][b[n, Join[l[[1;; k-1]],
       Table[1+i-k, {j, k, i}], l[[i+1;;]]]]]]; s]];
a[n_] := b[n, Array[0&, n]][[2]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz in A226545 *)

a(16) from Alois P. Heinz, Nov 16 2016

cp's OEIS Frontend

A063443 Number of ways to tile an n X n square with 1 X 1 and 2 X 2 tiles.

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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.

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A219924 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A224239 Number of inequivalent ways to cut an n X n square into squares with integer sides.

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A226979 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 2 elements.

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A226980 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

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A225777 Number T(n,k,u) of distinct tilings of an n X k rectangle using integer-sided square tiles containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k <= n, u >= 0, read by rows.

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A226978 Number of ways to cut an n X n square into squares with integer sides, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element.

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