A211348 -id:A211348 - 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

A045846 Number of distinct ways to cut an n X n square into squares with integer sides.

Original entry on oeis.org

1, 1, 2, 6, 40, 472, 10668, 450924, 35863972, 5353011036, 1500957422222, 790347882174804, 781621363452405930, 1451740730942350766748, 5064070747064013556294032, 33176273260130056822126522884, 408199838581532754602910469192704

Offset: 0

Erich Friedman

			For n=3 the 6 dissections are: the full 3 X 3 square; 9 1 X 1 squares; one 2 X 2 square and five 1 X 1 squares (in 4 ways).

Andrew Gozzard and Max Ward, Table of n, a(n) for n = 0..25 (terms 0..20 from Steve Butler).
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.
N. J. A. Sloane, Illustration of the first five terms of A045846 and A224239, page 1 of 4 (Each dissection from A224239 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 of A045846 and A224239, 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 of A045846 and A224239, 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 of A045846 and A224239, 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, arXiv:1308.5420 [math.CO], 2013-2014.

Diagonal of A219924. - Alois P. Heinz, Dec 01 2012
See A224239 for the number of inequivalent ways.
Cf. A034295, A063443, A211348, A226554.

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]$j=i+1..nops(l)])
         od; s
      fi
    end:
a:= n-> b(n, [0$n]):
seq(a(n), n=0..11);  # Alois P. Heinz, Apr 15 2013

$RecursionLimit = 1000; 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, For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; 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, {i-k+1}], l[[i+1 ;; Length[l]]]]]]; s]]; a[n_] := b[n, Array[0&, n]]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

It appears lim n->oo a(n)*a(n-3)/(a(n-1)*a(n-2)) = 3.527... - Gerald McGarvey, May 03 2005
It appears that lim n->oo a(n)*a(n-2)/(a(n-1))^2 = 1.8781... - Christopher Hunt Gribble, Jun 21 2013
a(n) = (1/n^2) * Sum_{k=1..n} k^2 * A226936(n,k). - Alois P. Heinz, Jun 22 2013

More terms from Hugo van der Sanden, Nov 06 2000
a(14)-a(15) from Alois P. Heinz, Nov 30 2012
a(16) from Steve Butler, Mar 14 2014

A335560 Number of ways to tile an n X n square with 1 X 1 squares and (n-1) X 1 vertical or horizontal strips.

Original entry on oeis.org

1, 16, 131, 335, 851, 2207, 5891, 16175, 45491, 130367, 378851, 1112015, 3286931, 9762527, 29091011, 86879855, 259853171, 777986687, 2330814371, 6986151695, 20945872211, 62812450847, 188387020931, 565060399535, 1694979872051, 5084536963007, 15252805582691

Offset: 1

Oluwatobi Jemima Alabi, Jun 14 2020

It is assumed that 1 X 1 squares and 1 X 1 strips can be distinguished. - Alois P. Heinz, Feb 23 2022

			Here is one of the 131 ways to tile a 3 X 3 square, in this case using two horizontal and two vertical strips:
   _ _ _
  |_ _| |
  | |_|_|
  |_|_ _|

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A063443 and A211348 (tiling an n X n square with smaller squares).

Cf. A028420 (tiling an n X n square with monomers and dimers).

Join[{1, 16}, LinearRecurrence[{6, -11, 6}, {131, 335, 851}, 25]] (* Amiram Eldar, Jun 16 2020 *)

Vec(x*(1 + 10*x + 46*x^2 - 281*x^3 + 186*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Jun 14 2020

a(n) = 2*3^n + 12*2^n - 19, for n >= 3.
From Colin Barker, Jun 14 2020: (Start)
G.f.: x*(1 + 10*x + 46*x^2 - 281*x^3 + 186*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>5. (End)
E.g.f.: 5 - 19*exp(x) +12 *exp(2*x) + 2*exp(3*x) - 10*x - 31*x^2/2. - Stefano Spezia, Aug 25 2025

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

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A045846 Number of distinct ways to cut an n X n square into squares with integer sides.

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A335560 Number of ways to tile an n X n square with 1 X 1 squares and (n-1) X 1 vertical or horizontal strips.

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