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 A330805 #29 Sep 19 2020 18:06:59
%S A330805 0,9,51,166,410,855,1589,2716,4356,6645,9735,13794,19006,25571,33705,
%T A330805 43640,55624,69921,86811,106590,129570,156079,186461,221076,260300,
%U A330805 304525,354159,409626,471366,539835,615505,698864,790416,890681,1000195,1119510,1249194,1389831
%N A330805 Number of squares and rectangles in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.
%C A330805 Collection: 2*n*(n+1)-ominoes.
%C A330805 Number of squares (all sizes): (8*n^3 + 24*n^2 + 22*n - 3*(-1)^n + 3)/12.
%C A330805 Number of rectangles (all sizes): (8*n^4 + 24*n^3 + 22*n^2 + 3*(-1)^n - 3)/12.
%H A330805 Teofil Bogdan and Mircea Dan Rus, <a href="https://arxiv.org/abs/2007.13472">Counting the lattice rectangles inside Aztec diamonds and square biscuits</a>, arXiv:2007.13472 [math.CO], 2020.
%H A330805 Luce ETIENNE, <a href="/A330805/a330805.pdf">Illustration of a(1), a(2) and a(3)</a>.
%H A330805 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A330805 G.f.: x*(x + 3)^2/(1 - x)^5.
%F A330805 E.g.f.: (1/6)*exp(x)*x*(54 + 99*x + 40*x^2 + 4*x^3). - _Stefano Spezia_, Jan 01 2020
%F A330805 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A330805 a(n) = n*(n + 1)*(4*n^2 + 12*n + 11)/6.
%F A330805 a(n) = 4*A000332(n+3) + A212523(n+1).
%F A330805 a(n) = 9*A000332(n+3) + 6*A000332(n+2) + A000332(n+1). - _Mircea Dan Rus_, Aug 26 2020
%F A330805 a(n) = 3*A004320(n) + A004320(n-1). - _Mircea Dan Rus_, Aug 26 2020
%e A330805 a(1) = 4*1+5 = 9; a(2) = 4*5+31 = 51; a(3) = 4*15 + 106 = 166; a(4) = 4*36 + 270 = 410.
%t A330805 LinearRecurrence[{5,-10,10,-5,1},{0,9,51,166,410},40] (* _Harvey P. Dale_, Jun 27 2020 *)
%Y A330805 Cf. A000332, A004320, A046092, A111746, A212523.
%K A330805 nonn,easy
%O A330805 0,2
%A A330805 _Luce ETIENNE_, Jan 01 2020