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 A348919 #13 Nov 19 2021 16:12:25
%S A348919 0,0,0,1,2,5,10,18,29,47,69,100,140,191,253,333,426,540,675,834,1017,
%T A348919 1234,1478,1760,2080,2442,2846,3305,3810,4375,5000,5690,6445,7281,
%U A348919 8187,9180,10260,11433,12699,14077,15554,17150,18865,20706,22673,24788,27036,29440,32000,34724
%N A348919 Sum of the middle parts of the partitions of k into 3 parts for all 0 <= k <= n.
%C A348919 Partial sums of A308266.
%H A348919 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A348919 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,0,-3,-3,3,3,0,-2,-1,1).
%F A348919 a(n) = Sum_{m=1..n} Sum_{k=1..floor(m/3)} Sum_{i=k..floor((m-k)/2)} i.
%F A348919 G.f.: -x^3*(x^4+x^3+x^2+x+1)/((x+1)^2*(x^2+x+1)^2*(x-1)^5). - _Alois P. Heinz_, Nov 03 2021
%F A348919 a(n) ~ 5*n^4/864. - _Stefano Spezia_, Nov 04 2021
%F A348919 a(n) = a(n-1)+2*a(n-2)-3*a(n-4)-3*a(n-5)+3*a(n-6)+3*a(n-7)-2*a(n-9)-a(n-10)+a(n-11). - _Wesley Ivan Hurt_, Nov 19 2021
%e A348919 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A348919 1+1+8
%e A348919 1+1+7 1+2+7
%e A348919 1+2+6 1+3+6
%e A348919 1+1+6 1+3+5 1+4+5
%e A348919 1+1+5 1+2+5 1+4+4 2+2+6
%e A348919 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e A348919 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e A348919 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 A348919 -----------------------------------------------------------------------
%e A348919 n | 3 4 5 6 7 8 9 10 ...
%e A348919 -----------------------------------------------------------------------
%e A348919 a(n) | 1 2 5 10 18 29 47 69 ...
%e A348919 -----------------------------------------------------------------------
%t A348919 CoefficientList[Series[-x^3*(x^4 + x^3 + x^2 + x + 1)/((x + 1)^2*(x^2 + x + 1)^2*(x - 1)^5), {x, 0, 49}], x] (* _Michael De Vlieger_, Nov 05 2021 *)
%Y A348919 Cf. A115266, A308266.
%K A348919 nonn,easy
%O A348919 0,5
%A A348919 _Wesley Ivan Hurt_, Nov 03 2021