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A270060 Number of incomplete rectangles of area n.

%I A270060 #41 Jul 13 2025 10:59:29
%S A270060 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 A270060 64,75,71,84,79,95,89,106,92,116,104,127,116,134,121,150,130,160,143,
%U A270060 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 A270060 Number of incomplete rectangles of area n.
%C A270060 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 A270060 In other words, this shape is a rectangle from which a smaller rectangle has been removed from one corner.
%C A270060 Incomplete rectangles which differ by a rotation and/or reflection are not counted as different.
%C A270060 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 A270060 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 A270060 a(n)=(A002133(n)+A100073(n))/2. See the integer partition comment above. _Lara Pudwell_, Apr 03 2016
%F A270060 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 A270060 n = 3
%e A270060 .___.
%e A270060 | ._|
%e A270060 |_|
%e A270060 .
%e A270060 n = 4
%e A270060 ._____.
%e A270060 | .___|
%e A270060 |_|
%e A270060 .
%e A270060 n = 5
%e A270060 ._______. ._____. ._____.
%e A270060 | ._____| |   ._| | .___|
%e A270060 |_|       |___|   | |
%e A270060                   |_|
%e A270060 .
%e A270060 The three solutions for n = 6:
%e A270060 XXXXX
%e A270060 X
%e A270060 .....
%e A270060 XXXX
%e A270060 XX
%e A270060 .....
%e A270060 XXXX
%e A270060 X
%e A270060 X
%e A270060 .....
%p A270060 # see A067627(n,k=2).
%o A270060 (Pseudocode)
%o A270060 /* rectangle : LL = long side, SS = short side
%o A270060 removed corner : L = long side, S = short side */
%o A270060 {
%o A270060 int a[100];
%o A270060 int LL,SS,L,S,area;
%o A270060 for(area:=1;area<=100;area++){
%o A270060 a[area]:=0;
%o A270060 };
%o A270060 for(LL:=1;LL<=100;LL++){
%o A270060 for(SS:=1;SS<=LL;SS++){
%o A270060   for(L:=1;L<=LL;L++){
%o A270060     for(S:=1;S<=LL;S++){
%o A270060       area=LL*SS-L*S;
%o A270060       if( area>=1 && area<=100 ){
%o A270060         if( L>=S || L<LL || S<SS ){
%o A270060           a[area]++;
%o A270060         };
%o A270060         if( L<S || L<SS || S<LL || LL>SS ){
%o A270060           a[area]++;
%o A270060         };
%o A270060       };
%o A270060     };
%o A270060   };
%o A270060 };
%o A270060 };
%o A270060 for(area:=1;area<=100;area++){
%o A270060   print a[area];
%o A270060 };
%o A270060 }
%Y A270060 Cf. A038548 (number of complete rectangles of area n), A002133, A100073, A067627.
%K A270060 nonn
%O A270060 1,5
%A A270060 _Stanislav Mikusek_, Mar 09 2016