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 A226912 Irregular triangle T(n,k) is the frequency with which the number of square parts equals k in each partition of an n X n square lattice into squares, considering only the list of parts. 3
 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 3, 0, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,64 COMMENTS The sequence was derived from the documents in the Links section.  The documents are first specified in the Links section of A034295. The irregular triangle is presented below.   k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ... n 1    1 2    1  0  0  1 3    1  0  0  0  0  1  0  0  1 4    1  0  0  1  0  0  1  1  0  1  0  0  1  0  0  1 5    1  0  0  0  0  0  0  1  0  1  1  0  1  1  0  1  1  0  1  0 ... 6    1  0  0  1  0  1  0  0  3  0  1  4  1  1  2  1  1  2  1  1 ... 7    1  0  0  0  0  0  0  0  1  2  0  2  2  2  2  2  3  2  3  2 ... 8    1  0  0  1  0  0  1  1  0  3  3  2  5  3  3  8  5  4  7  6 ... LINKS Alois P. Heinz, Rows n = 1..13, flattened 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 It appears that for n > 6, T(n, floor(n^2/2) + 3 : n^2) =  T(n-1, floor(n^2/2) - 2n + 4 : (n-1)^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 So T(3,1) = 1, T(3,6) = 1, T(3,9) = 1. 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: T:= n-> (p-> seq(coeff(p, z, i), i=1..n^2))         (add(z^subs(x=1, f), f=b(n, [0\$n]))): seq(T(n), n=1..9);  # 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[# + x^(1+i-k)&, b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; Length[l]]]]]]]; s]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, n^2}]][Sum[z^(f /. x -> 1), {f, b[n, Array[0&, n]]}]]; Table[T[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 24 2016, after Alois P. Heinz *) CROSSREFS Row sums = A034295. Row lengths are A000290. Sequence in context: A178116 A238709 A245120 * A177330 A197126 A256987 Adjacent sequences:  A226909 A226910 A226911 * A226913 A226914 A226915 KEYWORD nonn,tabf,hard AUTHOR Christopher Hunt Gribble, Jun 22 2013 STATUS approved

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Last modified February 27 14:47 EST 2020. Contains 332306 sequences. (Running on oeis4.)