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A270060
Number of incomplete rectangles of area n.
6
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
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
.___.
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.
n = 4
._____.
| .___|
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.
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: A003879 A376821 A333885 * A350636 A078565 A211541
KEYWORD
nonn
AUTHOR
Stanislav Mikusek, Mar 09 2016
STATUS
approved