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A225622 A(n,k) is the total number of parts in the set of partitions of an n X k rectangle into integer sided squares, considering only the list of parts; square array A(n,k), n>=1, k>=1, read by antidiagonals. 4
1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 5, 15, 16, 15, 5, 6, 21, 31, 31, 21, 6, 7, 30, 47, 59, 47, 30, 7, 8, 38, 73, 102, 102, 73, 38, 8, 9, 50, 101, 170, 156, 170, 101, 50, 9, 10, 60, 142, 250, 307, 307, 250, 142, 60, 10, 11, 75, 185, 375, 460, 529, 460, 375, 185, 75, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The square array starts:

1    2    3    4    5    6    7    8    9   10   11   12 ...

2    5    9   15   21   30   38   50   60   75   87  105 ...

3    9   16   31   47   73  101  142  185  244  305  386 ...

4   15   31   59  102  170  250  375  523  726  962 ...

5   21   47  102  156  307  460  711 1040 1517 ...

6   30   73  170  307  529  907 1474 2204 ...

7   38  101  250  460  907 1351 2484 ...

8   50  142  375  711 1474 2484 ...

9   60  185  523 1040 2204 ...

LINKS

Christopher Hunt Gribble and Alois P. Heinz, Antidiagonals n = 1..23, flattened (Antidiagonals n = 1..14 from Christopher Hunt Gribble)

Christopher Hunt Gribble, C++ program

FORMULA

A(n,1) = A000027(n) = n.

A(n,2) = A195014(n) = (n+1)(5n+3)/8 when n is odd

                      and 5n(n+2)/8 when n is even.

EXAMPLE

A(3,2) = 9 because there are 9 parts overall in the 2 partitions of a 3 X 2 rectangle into squares with integer sides.  One partition comprises 6 1 X 1 squares and the other 2 1 X 1 squares and 1 2 X 2 square giving 9 parts in total.

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, k)-> add(coeff(add(j, j=b(max(n, k),

            [0$min(n, k)])), x, i), i=1..n):

seq(seq(A(n, 1+d-n), n=1..d), d=1..15); # Alois P. Heinz, Aug 04 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, For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; 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_, k_] := Sum[Coefficient[Sum[j, {j, b[Max[n, k], Array[0&, Min[n, k]]]}], x, i], {i, 1, n}]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 15}] // Flatten (* Jean-Fran├žois Alcover, Mar 06 2015, after Alois P. Heinz *)

CROSSREFS

Diagonal = A226897.

Cf. A034295, A224697, A225542.

Sequence in context: A128141 A252829 A014430 * A196436 A197199 A196957

Adjacent sequences:  A225619 A225620 A225621 * A225623 A225624 A225625

KEYWORD

nonn,tabl

AUTHOR

Christopher Hunt Gribble, Aug 04 2013

STATUS

approved

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Last modified June 27 11:25 EDT 2017. Contains 288788 sequences.