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%I #11 Jul 09 2020 19:38:43
%S 0,0,1,1,3,5,8,11,18,22,31,40,51,62,80,93,114,135,159,183,217,244,282,
%T 320,362,404,459,505,565,625,690,755,836,906,993,1080,1173,1266,1378,
%U 1477,1596,1715,1841,1967,2115,2248,2404,2560,2724,2888,3077,3249,3447
%N Sum of the middle 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)} i.
%F Conjectures from _Colin Barker_, Jul 16 2019: (Start)
%F G.f.: x^3*(1 + x + x^2 + x^3 + 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 1 3 5 8 11 18 22 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[i, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%t Table[Total[IntegerPartitions[n,{3}][[All,2]]],{n,60}] (* _Harvey P. Dale_, Jul 09 2020 *)
%Y Cf. A308265.
%K nonn
%O 1,5
%A _Wesley Ivan Hurt_, May 17 2019
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