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%I #11 Jan 16 2022 11:24:12
%S 0,0,0,3,2,4,4,8,8,13,12,18,18,24,24,33,32,40,40,50,50,61,60,72,72,84,
%T 84,99,98,112,112,128,128,145,144,162,162,180,180,201,200,220,220,242,
%U 242,265,264,288,288,312,312,339,338,364,364,392,392,421,420
%N Number of odd 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_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ((i mod 2) + (j mod 2) + ((n-i-j) mod 2)).
%F Conjectures from _Colin Barker_, Aug 06 2019: (Start)
%F G.f.: x^3*(3 - x + 2*x^2 + x^4 + x^5) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
%F a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) 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) | 3 2 4 4 8 8 13 12 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[Mod[i, 2] + Mod[j, 2] + Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t Table[Count[Flatten[IntegerPartitions[n,{3}]],_?OddQ],{n,0,60}] (* _Harvey P. Dale_, Jan 16 2022 *)
%K nonn
%O 0,4
%A _Wesley Ivan Hurt_, Aug 05 2019