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 A309694 #14 Sep 01 2019 11:25:19
%S A309694 0,0,0,0,2,2,6,4,14,14,28,24,48,44,74,68,112,106,158,144,214,206,286,
%T A309694 268,370,352,466,444,584,562,716,680,864,838,1038,996,1230,1188,1440,
%U A309694 1392,1682,1634,1944,1876,2228,2174,2548,2472,2892,2816,3260,3176,3670
%N A309694 Sum of the even parts appearing among the largest parts of the partitions of n into 3 parts.
%H A309694 Colin Barker, <a href="/A309694/b309694.txt">Table of n, a(n) for n = 0..1000</a>
%H A309694 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A309694 <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,1,-1,3,-3,2,-2,-2,2,-3,3,-1,1,1,-1,1,-1).
%F A309694 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (n-i-j) * ((n-i-j-1) mod 2).
%F A309694 From _Colin Barker_, Aug 23 2019: (Start)
%F A309694 G.f.: 2*x^4*(1 + 3*x^2 - x^3 + 6*x^4 - x^5 + 7*x^6 - x^7 + 6*x^8 - x^9 + 3*x^10 - x^11 + x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).
%F A309694 a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + 3*a(n-6) - 3*a(n-7) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14) + a(n-15) + a(n-16) - a(n-17) + a(n-18) - a(n-19) for n>18.
%F A309694 (End)
%e A309694 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A309694 1+1+8
%e A309694 1+1+7 1+2+7
%e A309694 1+2+6 1+3+6
%e A309694 1+1+6 1+3+5 1+4+5
%e A309694 1+1+5 1+2+5 1+4+4 2+2+6
%e A309694 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e A309694 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e A309694 1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
%e A309694 -----------------------------------------------------------------------
%e A309694 n | 3 4 5 6 7 8 9 10 ...
%e A309694 -----------------------------------------------------------------------
%e A309694 a(n) | 0 2 2 6 4 14 14 28 ...
%e A309694 -----------------------------------------------------------------------
%t A309694 Table[Sum[Sum[(n - i - j) * Mod[n - i - j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t A309694 LinearRecurrence[{1, -1, 1, 1, -1, 3, -3, 2, -2, -2, 2, -3, 3, -1, 1, 1, -1, 1, -1}, {0, 0, 0, 0, 2, 2, 6, 4, 14, 14, 28, 24, 48, 44, 74, 68, 112, 106, 158}, 80]
%o A309694 (PARI) concat([0,0,0,0], Vec(2*x^4*(1 + 3*x^2 - x^3 + 6*x^4 - x^5 + 7*x^6 - x^7 + 6*x^8 - x^9 + 3*x^10 - x^11 + x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^40))) \\ _Colin Barker_, Aug 23 2019
%Y A309694 Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692.
%K A309694 nonn,easy
%O A309694 0,5
%A A309694 _Wesley Ivan Hurt_, Aug 12 2019