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 A226937 Number of different numbers of square parts in the set of partitions of an n X n square lattice into squares, considering only the list of parts. 1
 1, 2, 3, 7, 11, 23, 34, 52, 68, 87, 105, 134, 153, 182, 213, 237 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The sequence was derived from the documents in the Links section.  The documents are first specified in the Links section of A034295. a(n) is the number of nonzero columns in the n-th row of the irregular triangle specified in A226912. 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 FORMULA a(n) <= n^2. EXAMPLE For n = 3, the partitions are: Square side 1 2 3 Number of parts             9 0 0       9             5 1 0       6             0 0 1       1 As the number of parts for each partition is different, a(3) = 3. 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+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-> nops(b(n, [0\$n])): seq(a(n), n=1..10);  # Alois P. Heinz, Jun 22 2013 MATHEMATICA 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, k <= Length[l], k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s  ~Union~ Map[# + 1 &, b[n, Join[ l[[1 ;; k - 1]], Array[ 1 + i - k &, i - k + 1], l[[i + 1 ;; Length[l] ]]]]]]; s]]; a[n_] := Length[b[n, Array[0&, n]]]; Table[an = a[n]; Print[ "a(", n, ") = ", an]; an, {n, 1, 16}] (* Jean-François Alcover, Jan 24 2016, after Alois P. Heinz *) CROSSREFS Cf. A034295, A226912. Sequence in context: A049091 A039787 A267503 * A227199 A129940 A128631 Adjacent sequences:  A226934 A226935 A226936 * A226938 A226939 A226940 KEYWORD nonn,more,hard AUTHOR Christopher Hunt Gribble, Jun 22 2013 EXTENSIONS a(14) from Alois P. Heinz, Jun 22 2013 Two more terms from Jean-François Alcover, Jan 24 2016 STATUS approved

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Last modified July 18 07:10 EDT 2019. Contains 325134 sequences. (Running on oeis4.)