A270060 Number of incomplete rectangles of area n.

0, 0, 1, 1, 3, 3, 6, 7, 9, 11, 14, 15, 19, 22, 23, 28, 30, 34, 36, 41, 42, 51, 49, 57, 55, 68, 64, 75, 71, 84, 79, 95, 89, 106, 92, 116, 104, 127, 116, 134, 121, 150, 130, 160, 143, 172, 148, 188, 156, 193, 177, 209, 177, 226, 185, 231, 210, 246, 207, 269, 218, 272, 239, 287, 238, 312, 250, 317, 279, 320, 271, 359, 283, 355, 316

Offset

1,5

Comments

An incomplete rectangle is a six-sided figure obtained when two rectangles with different widths are coupled together so that two of the edges form a straight line.

In other words, this shape is a rectangle from which a smaller rectangle has been removed from one corner.

Incomplete rectangles which differ by a rotation and/or reflection are not counted as different.

Also the number of integer partitions of n into parts of 2 distinct sizes, where any integer partition and its conjugate are considered equivalent. For example a(8)=7 counts (7,1), (6,2), (6,1,1), (5,3), (5,1,1,1), (4,2,2), and (3,3,2).

The unit squares composing the incomplete rectangle can be viewed as the boxes of a Ferrers diagram of an integer partition of n with 2 different sizes of rows. A002133(n) counts all Ferrers diagrams with 2 different sizes of rows. A100073(n) counts all self-conjugate Ferrers diagrams with 2 different sizes of rows since these Ferrers diagrams look like a square with a smaller square removed from the corner. Thus a(n)=(A002133(n)+A100073(n))/2. Lara Pudwell, Apr 03, 2016

Links

Table of n, a(n) for n=1..75.

Formula

a(n)=(A002133(n)+A100073(n))/2. See the integer partition comment above. Lara Pudwell, Apr 03, 2016

G.f.: sum(sum(x^(i+j)/(2*(1-x^i)*(1-x^j))+x^(i^2-j^2)/2,j=1..i-1),i=1..infinity). See the integer partition comment above. Lara Pudwell, Apr 03, 2016

Example

n = 3
.___.
| ._|
|_|
.
n = 4
._____.
| .___|
|_|
.
n = 5
._______. ._____. ._____.
| ._____| |   ._| | .___|
|_|       |___|   | |
                  |_|
.
The three solutions for n = 6:
XXXXX
X
.....
XXXX
XX
.....
XXXX
X
X
.....

Maple

# see A067627(n, k=2).

Prog

(C)
/* rectangle : LL = long side, SS = short side
removed corner : L = long side, S = short side */
{
int a[100];
int LL, SS, L, S, area;
for(area:=1; area<=100; area++){
a[area]:=0;
};
for(LL:=1; LL<=100; LL++){
for(SS:=1; SS<=LL; SS++){
  for(L:=1; L<=LL; L++){
    for(S:=1; S<=LL; S++){
      area=LL*SS-L*S;
      if( area>=1 && area<=100 ){
        if( L>=S || L<LL || S<SS ){
          a[area]++;
        };
        if( L<S || L<SS || S<LL || LL>SS ){
          a[area]++;
        };
      };
    };
  };
};
};
for(area:=1; area<=100; area++){
  print a[area];
};
}

Crossrefs

Cf. A038548 (number of complete rectangles of area n), A002133, A100073, A067627.

Sequence in context: A269170 A003879 A333885 * A350636 A078565 A211541

Adjacent sequences: A270057 A270058 A270059 * A270061 A270062 A270063

Keyword

nonn

Author

Stanislav Mikusek, Mar 09 2016

Status

approved