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%I #19 Sep 02 2019 07:37:42
%S 0,0,0,1,1,2,2,3,3,5,5,7,7,9,9,12,12,15,15,18,18,22,22,26,26,30,30,35,
%T 35,40,40,45,45,51,51,57,57,63,63,70,70,77,77,84,84,92,92,100,100,108,
%U 108,117,117,126,126,135,135,145,145,155,155,165,165,176
%N Number of odd parts appearing among the smallest parts of the partitions of n into 3 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,1,-1,-1,1).
%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (j mod 2).
%F From _Colin Barker_, Aug 22 2019: (Start)
%F G.f.: x^3 / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)).
%F a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) for n>8.
%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 2 2 3 3 5 5 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[Mod[j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 0, 0, 1, 1, 2, 2, 3, 3}, 50] (* _Wesley Ivan Hurt_, Aug 28 2019 *)
%Y Cf. A026923, A026927, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.
%K nonn,easy
%O 0,6
%A _Wesley Ivan Hurt_, Aug 12 2019
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