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 A317714 #44 Jul 04 2025 03:08:48
%S A317714 0,0,1,2,10,18,45,72,136,200,325,450,666,882,1225,1568,2080,2592,3321,
%T A317714 4050,5050,6050,7381,8712,10440,12168,14365,16562,19306,22050,25425,
%U A317714 28800,32896,36992,41905,46818,52650,58482,65341,72200,80200,88200,97461,106722,117370
%N A317714 Chessboard rectangles sequence (see Comments), also A037270 interleaved with A163102.
%C A317714 Take a chessboard of n X n unit squares in which the a1 square is black. a(n) is the number of composite rectangles of p X q unit squares whose vertices are covered by black unit squares (1 < p <= n, 1 < q <= n).
%H A317714 Jinyuan Wang, <a href="/A317714/b317714.txt">Table of n, a(n) for n = 1..10000</a>
%H A317714 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A317714 a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8), with a(1)=0, a(2)=0, a(3)=1, a(4)=2, a(5)=10, a(6)=18, a(7)=45, a(8)=72.
%F A317714 G.f.: -(x^3*(1+ 4*x^2 + x^4))/((-1+x)^5*(1+x)^3).
%F A317714 a(n) = (5 - 5*(-1)^n - 12*n + 12*(-1)^n*n + 14*n^2 - 6*(-1)^n*n^2 - 8*n^3 + 2*n^4)/64.
%F A317714 a(n) = Sum_{i=1..n-1} floor(i/2)^3. - _Ridouane Oudra_, Jul 24 2019
%F A317714 E.g.f.: (1/64)*exp(-x)*(-5-6*x-6*x^2+exp(2*x)*(5-4*x+4*x^2+4*x^3+2*x^4)). - _Stefano Spezia_, Aug 14 2019
%F A317714 a(2*n) = A163102(n-1) and a(2*n+1) = A037270(n). - _Ridouane Oudra_, Mar 24 2024
%F A317714 Sum_{n>=3} 1/a(n) = Pi^2 - Pi*coth(Pi) - 5. - _Amiram Eldar_, Jul 04 2025
%e A317714 In a 4 X 4 chessboard there are two such rectangles (for both p = q = 3) and the coordinates of their lower left vertices are a1 and b2. Therefore, a(4) = 2.
%t A317714 CoefficientList[Series[-((x^2 (1+4 x^2+x^4))/((-1+x)^5 (1+x)^3)),{x,0,44}],x]
%t A317714 LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 1, 2, 10, 18, 45, 72}, 80] (* _Vincenzo Librandi_, Aug 06 2018 *)
%o A317714 (Magma) [(5-5*(-1)^n-12*n+12*(-1)^n*n+14*n^2-6*(-1)^n*n^2-8*n^3+2*n^4)/64: n in [1..50]]; // _Vincenzo Librandi_, Aug 05 2018
%o A317714 (Python)
%o A317714 n, a = 0, 0
%o A317714 while n < 10:
%o A317714 print(n,a)
%o A317714 n, a = n+1, a+((n+1)//2)**3 # _A.H.M. Smeets_, Aug 09 2019
%o A317714 (PARI) a(n) = sum(i = 1, n-1, floor(i/2)^3); \\ _Jinyuan Wang_, Aug 12 2019
%Y A317714 Cf. A005993, A037270, A163102.
%K A317714 easy,nonn
%O A317714 1,4
%A A317714 _Ivan N. Ianakiev_, Aug 05 2018