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 A308823 #15 Nov 20 2024 18:57:10
%S A308823 0,0,0,0,0,1,1,2,3,5,8,11,15,21,28,38,48,62,78,98,122,149,181,219,262,
%T A308823 314,370,436,510,595,691,797,916,1050,1198,1365,1545,1747,1968,2212,
%U A308823 2480,2771,3089,3437,3814,4227,4669,5151,5670,6232,6838,7487,8185,8936
%N A308823 Sum of the smallest parts of the partitions of n into 5 parts.
%H A308823 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A308823 a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} l.
%F A308823 a(n) = A308822(n) - A308824(n) - A308825(n) - A308826(n) - A308827(n).
%F A308823 Conjectures from _Colin Barker_, Jun 30 2019: (Start)
%F A308823 G.f.: x^5 / ((1 - x)^6*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)^2).
%F A308823 a(n) = a(n-1) + a(n-2) - 2*a(n-6) - 2*a(n-7) + a(n-8) + a(n-9) + 2*a(n-10) + a(n-11) + a(n-12) - 2*a(n-13) - 2*a(n-14) + a(n-18) + a(n-19) - a(n-20) for n>19.
%F A308823 (End)
%e A308823 Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
%e A308823 1+1+1+1+10
%e A308823 1+1+1+2+9
%e A308823 1+1+1+3+8
%e A308823 1+1+1+4+7
%e A308823 1+1+1+5+6
%e A308823 1+1+1+1+9 1+1+2+2+8
%e A308823 1+1+1+2+8 1+1+2+3+7
%e A308823 1+1+1+3+7 1+1+2+4+6
%e A308823 1+1+1+4+6 1+1+2+5+5
%e A308823 1+1+1+5+5 1+1+3+3+6
%e A308823 1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
%e A308823 1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
%e A308823 1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
%e A308823 1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
%e A308823 1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
%e A308823 1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
%e A308823 1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
%e A308823 1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
%e A308823 1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
%e A308823 1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
%e A308823 1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
%e A308823 1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
%e A308823 2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
%e A308823 --------------------------------------------------------------------------
%e A308823 n | 10 11 12 13 14 ...
%e A308823 --------------------------------------------------------------------------
%e A308823 a(n) | 8 11 15 21 28 ...
%e A308823 --------------------------------------------------------------------------
%e A308823 - _Wesley Ivan Hurt_, Sep 08 2019
%t A308823 Table[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]
%t A308823 Table[Total[IntegerPartitions[n,{5}][[;;,5]]],{n,0,60}] (* _Harvey P. Dale_, Nov 20 2024 *)
%Y A308823 Cf. A026811, A308822, A308824, A308825, A308826, A308827.
%K A308823 nonn
%O A308823 0,8
%A A308823 _Wesley Ivan Hurt_, Jun 26 2019