This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A303120 #24 Sep 08 2022 08:46:21
%S A303120 0,7,60,372,1300,4047,9800,22352,44280,84575,147620,251412,402220,
%T A303120 632247,949200,1406272,2011440,2847447,3920460,5353300,7147140,
%U A303120 9477567,12336280,15966672,20345000,25800047,32284980,40234292,49568540,60851175,73958560,89609472
%N A303120 Total area of all rectangles of size p X q such that p + q = n^2 and p <= q.
%C A303120 Sum of all the products formed using the corresponding largest and smallest parts of each partition of n^2 into two parts. - _Wesley Ivan Hurt_, Mar 26 2019
%H A303120 Muniru A Asiru, <a href="/A303120/b303120.txt">Table of n, a(n) for n = 1..2000</a>
%H A303120 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A303120 a(n) = Sum_{i=1..floor(n^2/2)} i * (n^2 - i).
%F A303120 Conjectures from _Colin Barker_, Apr 19 2018 and Mar 19 2019: (Start)
%F A303120 G.f.: x^2*(7 + 46*x + 224*x^2 + 386*x^3 + 594*x^4 + 386*x^5 + 224*x^6 + 46*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)^5).
%F A303120 a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n > 12.
%F A303120 a(n) = (n^2*(-4 + 3*(1+(-1)^n)*n^2 + 4*n^4)) / 48.
%F A303120 (End)
%e A303120 a(3) = 60; The rectangles are 8 X 1, 7 X 2, 6 X 3 and 5 X 4. The total area is then 8*1 + 7*2 + 6*3 + 5*4 = 60.
%e A303120 a(4) = 372; The rectangles are 15 X 1, 14 X 2, 13 X 3, 12 X 4, 11 X 5, 10 X 6, 9 X 7 and 8 X 8. The total area of the rectangles is then 15*1 + 14*2 + 13*3 + 12*4 + 11*5 + 10*6 + 9*7 + 8*8 = 372.
%p A303120 A303120:=n->add(i*(n^2-i), i=1..floor(n^2/2)): seq(A303120(n), n=1..50); # _Wesley Ivan Hurt_, Mar 12 2019
%t A303120 Table[Sum[i*(n^2 - i), {i, Floor[n^2/2]}], {n, 50}]
%o A303120 (Magma) [&+[i*(n^2-i): i in [0..Floor(n^2/2)]]: n in [1..35]]; // _Vincenzo Librandi_, Apr 19 2018
%o A303120 (PARI) a(n) = sum(i=1, n^2\2, i*(n^2-i)); \\ _Michel Marcus_, Mar 13 2019
%o A303120 (GAP) List([1..35],n->Sum([1..Int(n^2/2)],i->i*(n^2-i))); # _Muniru A Asiru_, Mar 15 2019
%Y A303120 Cf. A000290, A023855.
%K A303120 nonn,easy
%O A303120 1,2
%A A303120 _Wesley Ivan Hurt_, Apr 18 2018