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

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

Offset

0,3

Links

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.

Formula

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

Example

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).

Maple

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

Mathematica

$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 *)

Crossrefs

Diagonal of A219924. - Alois P. Heinz, Dec 01 2012

See A224239 for the number of inequivalent ways.

Cf. A034295, A063443, A211348, A226554.

Sequence in context: A274275 A081471 A133939 * A238818 A199574 A320489

Adjacent sequences: A045843 A045844 A045845 * A045847 A045848 A045849

Keyword

hard,nonn,nice

Author

Erich Friedman

Extensions

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

Status

approved