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 A309690 #14 Sep 03 2019 23:23:58
%S A309690 0,0,0,0,0,2,4,4,4,8,12,16,20,26,32,38,44,58,72,80,88,106,124,142,160,
%T A309690 182,204,226,248,284,320,346,372,414,456,498,540,588,636,684,732,800,
%U A309690 868,922,976,1052,1128,1204,1280,1364,1448,1532,1616,1726,1836,1928
%N A309690 Sum of the even parts appearing among the second largest parts of the partitions of n into 3 parts.
%H A309690 Colin Barker, <a href="/A309690/b309690.txt">Table of n, a(n) for n = 0..1000</a>
%H A309690 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A309690 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).
%F A309690 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} i * ((i-1) mod 2).
%F A309690 From _Colin Barker_, Aug 23 2019: (Start)
%F A309690 G.f.: 2*x^5*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).
%F A309690 a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 3*a(n-4) + 2*a(n-5) + a(n-6) - 4*a(n-7) + 6*a(n-8) - 8*a(n-9) + 6*a(n-10) - 4*a(n-11) + a(n-12) + 2*a(n-13) - 3*a(n-14) + 4*a(n-15) - 3*a(n-16) + 2*a(n-17) - a(n-18) for n>17.
%F A309690 (End)
%e A309690 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A309690 1+1+8
%e A309690 1+1+7 1+2+7
%e A309690 1+2+6 1+3+6
%e A309690 1+1+6 1+3+5 1+4+5
%e A309690 1+1+5 1+2+5 1+4+4 2+2+6
%e A309690 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e A309690 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e A309690 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 A309690 -----------------------------------------------------------------------
%e A309690 n | 3 4 5 6 7 8 9 10 ...
%e A309690 -----------------------------------------------------------------------
%e A309690 a(n) | 0 0 2 4 4 4 8 12 ...
%e A309690 -----------------------------------------------------------------------
%t A309690 Table[Sum[Sum[i * Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t A309690 LinearRecurrence[{2, -3, 4, -3, 2, 1, -4, 6, -8, 6, -4, 1, 2, -3, 4, -3, 2, -1}, {0, 0, 0, 0, 0, 2, 4, 4, 4, 8, 12, 16, 20, 26, 32, 38, 44, 58}, 80]
%o A309690 (PARI) concat([0,0,0,0,0], Vec(2*x^5*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^60))) \\ _Colin Barker_, Aug 23 2019
%Y A309690 Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309692, A309694.
%K A309690 nonn,easy
%O A309690 0,6
%A A309690 _Wesley Ivan Hurt_, Aug 12 2019