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%I #35 Nov 02 2021 06:32:33
%S 0,0,0,1,1,2,2,3,3,7,7,11,11,15,15,24,24,33,33,42,42,58,58,74,74,90,
%T 90,115,115,140,140,165,165,201,201,237,237,273,273,322,322,371,371,
%U 420,420,484,484,548,548,612,612,693,693,774,774,855,855,955,955
%N Sum of the odd parts appearing among the smallest parts of the partitions of n into 3 parts.
%H Jinyuan Wang, <a href="/A309684/b309684.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,2,-2,-2,2,0,0,-1,1,1,-1).
%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} j * (j mod 2).
%F From _Colin Barker_, Aug 22 2019: (Start)
%F G.f.: x^3*(1 + x^2)*(1 - x^2 + x^4) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x + x^2)^2).
%F a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-6) - 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-12) + a(n-13) + a(n-14) - a(n-15) for n > 14.
%F (End)
%F a(n) = (-4*s^3+(2*t-7)*s^2+(4*t-1)*s+2*t+2)/2, where s = floor((n-3)/6) and t = floor((n-3)/2). - _Wesley Ivan Hurt_, Oct 27 2021
%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 7 7 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[j*Mod[j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t LinearRecurrence[{1, 1, -1, 0, 0, 2, -2, -2, 2, 0, 0, -1, 1, 1, -1}, {0, 0, 0, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 15, 15}, 20] (* _Wesley Ivan Hurt_, Aug 29 2019 *)
%o (PARI) a(n) = sum(j = 1, floor(n/3), sum(i = j, floor((n-j)/2), j * (j%2))); \\ _Jinyuan Wang_, Aug 29 2019
%Y Cf. A026923, A026927, A309683, 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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