A226897 a(n) is the total number of parts in the set of partitions of an n X n square lattice into squares, considering only the list of parts.

1, 5, 16, 59, 156, 529, 1351, 3988, 10236, 27746, 66763, 176783, 412450

Offset: 1

Author

Christopher Hunt Gribble, Jun 21 2013

Keywords

• nonn
• hard
• more

Comments

The sequence was derived from the documents in the Links section. The documents are first specified in the Links section of A034295.

Examples

			For n = 3, the partitions are:
Square side 1 2 3 Total Parts
            9 0 0     9
            5 1 0     6
            0 0 1     1
Total                16
So a(3) = 16.
		

Links

• Jon E. Schoenfield, Table of solutions for n <= 12
• Alois P. Heinz, More ways to divide an 11 X 11 square into sub-squares
• Alois P. Heinz, List of different ways to divide a 13 X 13 square into sub-squares

Crossrefs

Cf. A034295, A045846, A226554.

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]$j=i+1..nops(l)]))
         od; s
      fi
    end:
a:= n-> add(coeff(add(j, j=b(n, [0$n])), x, i), i=1..n):
seq(a(n), n=1..9);  # Alois P. Heinz, Jun 21 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, k = Position[l, 0, 1, 1][[1, 1]]; 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_] := Sum[Coefficient[Sum[j, {j, b[n, Array[0&, n]]}], x, i], {i, 1, n}]; Table[a[n], {n, 1, 9}] (* Jean-François Alcover, May 29 2015, after Alois P. Heinz *)