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 A280186 #50 Aug 19 2018 16:27:12
%S A280186 0,0,0,0,2,4,10,16,28,40,60,80,110,140,182,224,280,336,408,480,570,
%T A280186 660,770,880,1012,1144,1300,1456,1638,1820,2030,2240,2480,2720,2992,
%U A280186 3264,3570,3876,4218,4560,4940,5320,5740,6160,6622,7084,7590,8096,8648,9200
%N A280186 Number of 3-element subsets of S = {1..n} whose sum is odd.
%C A280186 The same as A006584 (apart from the offset). - _R. J. Mathar_, Jan 15 2017
%C A280186 There are two cases: n is odd and n is even.
%C A280186 Let n be an odd integer and n > 3, the sum of 3 integers is odd when all of them are odd or one is odd and the others are even. Number of ways to choose 3 odd numbers: C((n+1)/2, 3). Number of ways to choose 2 even numbers and 1 odd: C((n-1)/2, 2)*C((n+1)/2, 1). Total number of ways: C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1).
%C A280186 Let n be an even integer and n > 3. Number of ways to choose 3 odd numbers: C(n/2, 3). Number of ways to choose 2 even numbers and 1 odd: C(n/2, 2)*C(n/2, 1). Total number of ways: C(n/2, 3) + C(n/2, 2)*C(n/2, 1).
%C A280186 Take a chessboard of n X n unit squares in which the a1 square is black. a(n) is the number of composite squares having white unit squares on their vertices. For the number of composite squares having black unit squares on their vertices see A005993. - _Ivan N. Ianakiev_, Aug 19 2018
%H A280186 Colin Barker, <a href="/A280186/b280186.txt">Table of n, a(n) for n = 0..1000</a>
%H A280186 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A280186 a(n) = C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1) when n is odd.
%F A280186 a(n) = C(n/2, 3) + C(n/2, 2)*C(n/2, 1) when n is even.
%F A280186 From _Colin Barker_, Dec 28 2016: (Start)
%F A280186 a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>5.
%F A280186 a(n) = n*(n - 1)*(n - 2)/12 for n even.
%F A280186 a(n) = (n - 1)*(n + 1)*(n - 3)/12 for n odd.
%F A280186 G.f.: 2*x^4 / ((1-x)^4*(1+x)^2). (End)
%F A280186 a(n) = ((-1)^n)*(-1+n)*(3 - 3*(-1)^n - 4*((-1)^n)*n + 2*((-1)^n)*n^2)/24. - _Ivan N. Ianakiev_, Aug 19 2018
%e A280186 For n = 5 then a(5) = 4. The subsets are: {1, 2, 4}, {1, 3, 5}, {2, 3, 4}, {2, 4, 5}.
%t A280186 Table[Binomial[(n + #)/2, 3] + Binomial[(n - #)/2, 2] Binomial[(n + #)/2, 1] &@ Boole@ OddQ@ n, {n, 0, 49}] (* or *)
%t A280186 CoefficientList[Series[2 x^4/((1 - x)^4*(1 + x)^2), {x, 0, 49}], x] (* _Michael De Vlieger_, Jan 07 2017 *)
%o A280186 (PARI) concat(vector(4), Vec(2*x^4 / ((1-x)^4*(1+x)^2) + O(x^60))) \\ _Colin Barker_, Dec 28 2016
%Y A280186 Essentially 2*A006918.
%Y A280186 Same as A006584.
%K A280186 nonn,easy
%O A280186 0,5
%A A280186 _Necip Fazil Patat_, Dec 28 2016
%E A280186 More terms from _Colin Barker_, Dec 28 2016