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%I #9 Jun 15 2020 23:19:25
%S 0,0,1,2,5,9,15,22,34,45,62,81,104,129,163,195,237,282,333,387,454,
%T 518,596,678,768,862,973,1080,1205,1335,1475,1620,1786,1947,2130,2319,
%U 2520,2727,2959,3185,3437,3696,3969,4249,4558,4860,5192,5532,5888,6252
%N Sum of the largest parts in the partitions of n into 3 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (n-i-k).
%F Conjectures from _Colin Barker_, Jul 16 2019: (Start)
%F G.f.: x^3*(1 + 2*x + 3*x^2 + 3*x^3 + 2*x^4) / ((1 - x)^4*(1 + x)^2*(1 + x + x^2)^2).
%F a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) - a(n-6) + 2*a(n-7) + 2*a(n-8) - a(n-10) for n>10.
%F (End)
%e Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e 1+1+8
%e 1+1+7 1+2+7
%e 1+2+6 1+3+6
%e 1+1+6 1+3+5 1+4+5
%e 1+1+5 1+2+5 1+4+4 2+2+6
%e 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e 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 -----------------------------------------------------------------------
%e n | 3 4 5 6 7 8 9 10 ...
%e -----------------------------------------------------------------------
%e a(n) | 1 2 5 9 15 22 34 45 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[n - i - k, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%Y Cf. A307872.
%K nonn
%O 1,4
%A _Wesley Ivan Hurt_, May 17 2019
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