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 A221640 Number T(n,k) of different numbers of square parts in the set of partitions of an n X k rectangle into squares with integer sides, considering only the list of parts; triangle T(n,k), 1<=k<=n, read by rows. 1
 1, 1, 2, 1, 2, 3, 1, 3, 4, 7, 1, 3, 5, 9, 11, 1, 4, 7, 12, 18, 23, 1, 4, 8, 15, 23, 30, 34, 1, 5, 10, 20, 27, 37, 43, 52, 1, 5, 12, 22, 32, 42, 50, 58, 68, 1, 6, 14, 27, 36, 47, 57, 68, 76, 87, 1, 6, 16, 30, 42, 54, 64, 75, 85, 96, 105 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The triangle begins: . k  1    2    3    4    5    6    7    8 n 1    1 2    1    2 3    1    2    3 4    1    3    4    7 5    1    3    5    9   11 6    1    4    7   12   18   23 7    1    4    8   15   23   30   34 8    1    5   10   20   27   37   43   52 LINKS Alois P. Heinz, Rows n = 1..14, flattened Christopher Hunt Gribble, C++ program EXAMPLE T(4,3) = 4 because there are 4 partitions of a 4 X 3 rectangle into integer sided squares with different numbers of parts: Partition                           Number of parts 12 1 X 1 squares                           12 .8 1 X 1 squares, 1 2 X 2 square            9 .4 1 X 1 squares, 2 2 X 2 squares           6 .3 1 X 1 squares, 1 3 X 3 square            4 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: T:= (n, k)-> nops(b(max(n, k), [0\$min(n, k)])): seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 08 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]]; T[n_, k_] := Length[b[Max[n, k], Array[0&, Min[n, k]]]]; Table[Table[ T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 24 2016, after Alois P. Heinz *) CROSSREFS Diagonal = A226937. Cf. A224697, A227998. Sequence in context: A330661 A091438 A011794 * A073300 A104468 A293003 Adjacent sequences:  A221637 A221638 A221639 * A221641 A221642 A221643 KEYWORD nonn,tabl AUTHOR Christopher Hunt Gribble, Aug 08 2013 EXTENSIONS More terms from Alois P. Heinz, Aug 08 2013 STATUS approved

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Last modified May 10 14:22 EDT 2021. Contains 343770 sequences. (Running on oeis4.)