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 A309689 #24 Nov 02 2021 07:05:03
%S A309689 0,0,0,0,0,1,2,2,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,22,24,26,28,30,
%T A309689 32,35,38,40,42,45,48,51,54,57,60,63,66,70,74,77,80,84,88,92,96,100,
%U A309689 104,108,112,117,122,126,130,135,140,145,150,155,160,165,170
%N A309689 Number of even parts appearing among the second largest parts of the partitions of n into 3 parts.
%H A309689 Colin Barker, <a href="/A309689/b309689.txt">Table of n, a(n) for n = 0..1000</a>
%H A309689 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A309689 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1,0,1,-2,2,-2,1).
%F A309689 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ((i-1) mod 2).
%F A309689 From _Colin Barker_, Aug 23 2019: (Start)
%F A309689 G.f.: x^5 / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
%F A309689 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) for n>9.
%F A309689 (End)
%F A309689 a(n) = (6*n^2+48*cos(n*Pi/3)-36*cos(n*Pi/2)+16*cos(2*n*Pi/3)-3*(-1)^n-25)/144. - _Ilya Gutkovskiy_, Oct 29 2021
%e A309689 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A309689 1+1+8
%e A309689 1+1+7 1+2+7
%e A309689 1+2+6 1+3+6
%e A309689 1+1+6 1+3+5 1+4+5
%e A309689 1+1+5 1+2+5 1+4+4 2+2+6
%e A309689 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e A309689 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e A309689 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 A309689 -----------------------------------------------------------------------
%e A309689 n | 3 4 5 6 7 8 9 10 ...
%e A309689 -----------------------------------------------------------------------
%e A309689 a(n) | 0 0 1 2 2 2 3 4 ...
%e A309689 -----------------------------------------------------------------------
%t A309689 Table[Sum[Sum[Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t A309689 LinearRecurrence[{2, -2, 2, -1, 0, 1, -2, 2, -2, 1}, {0, 0, 0, 0, 0, 1, 2, 2, 2, 3}, 80]
%o A309689 (PARI) concat([0,0,0,0,0], Vec(x^5 / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ _Colin Barker_, Aug 23 2019
%Y A309689 Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309690, A309692, A309694.
%K A309689 nonn,easy
%O A309689 0,7
%A A309689 _Wesley Ivan Hurt_, Aug 12 2019