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%I #34 May 08 2019 12:27:44
%S 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,
%T 64,75,71,84,79,95,89,106,92,116,104,127,116,134,121,150,130,160,143,
%U 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
%N Number of incomplete rectangles of area n.
%C 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.
%C In other words, this shape is a rectangle from which a smaller rectangle has been removed from one corner.
%C Incomplete rectangles which differ by a rotation and/or reflection are not counted as different.
%C 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).
%C 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
%F a(n)=(A002133(n)+A100073(n))/2. See the integer partition comment above. _Lara Pudwell_, Apr 03, 2016
%F 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
%e n = 3
%e .___.
%e | ._|
%e |_|
%e .
%e n = 4
%e ._____.
%e | .___|
%e |_|
%e .
%e n = 5
%e ._______. ._____. ._____.
%e | ._____| | ._| | .___|
%e |_| |___| | |
%e |_|
%e .
%e The three solutions for n = 6:
%e XXXXX
%e X
%e .....
%e XXXX
%e XX
%e .....
%e XXXX
%e X
%e X
%e .....
%p # see A067627(n,k=2).
%o (C)
%o /* rectangle : LL = long side, SS = short side
%o removed corner : L = long side, S = short side */
%o {
%o int a[100];
%o int LL,SS,L,S,area;
%o for(area:=1;area<=100;area++){
%o a[area]:=0;
%o };
%o for(LL:=1;LL<=100;LL++){
%o for(SS:=1;SS<=LL;SS++){
%o for(L:=1;L<=LL;L++){
%o for(S:=1;S<=LL;S++){
%o area=LL*SS-L*S;
%o if( area>=1 && area<=100 ){
%o if( L>=S || L<LL || S<SS ){
%o a[area]++;
%o };
%o if( L<S || L<SS || S<LL || LL>SS ){
%o a[area]++;
%o };
%o };
%o };
%o };
%o };
%o };
%o for(area:=1;area<=100;area++){
%o print a[area];
%o };
%o }
%Y Cf. A038548 (number of complete rectangles of area n), A002133, A100073, A067627.
%K nonn
%O 1,5
%A _Stanislav Mikusek_, Mar 09 2016