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%I #17 Sep 03 2019 23:03:16
%S 0,0,0,1,1,1,1,4,7,10,10,15,20,25,30,42,49,56,63,79,95,111,120,140,
%T 160,180,200,233,257,281,305,344,383,422,450,495,540,585,630,694,745,
%U 796,847,919,991,1063,1120,1200,1280,1360,1440,1545,1633,1721,1809
%N Sum of the odd parts appearing among the second largest 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_18">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).
%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} i * (i mod 2).
%F From _Colin Barker_, Aug 23 2019: (Start)
%F G.f.: x^3*(1 + x + x^2 + x^3 + x^4)*(1 - 2*x + 3*x^2 - 4*x^3 + 6*x^4 - 4*x^5 + 3*x^6 - 2*x^7 + x^8) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).
%F a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 3*a(n-4) + 2*a(n-5) + a(n-6) - 4*a(n-7) + 6*a(n-8) - 8*a(n-9) + 6*a(n-10) - 4*a(n-11) + a(n-12) + 2*a(n-13) - 3*a(n-14) + 4*a(n-15) - 3*a(n-16) + 2*a(n-17) - a(n-18) for n>17.
%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 1 1 4 7 10 10 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[i * Mod[i, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t LinearRecurrence[{2, -3, 4, -3, 2, 1, -4, 6, -8, 6, -4, 1, 2, -3, 4, -3, 2, -1}, {0, 0, 0, 1, 1, 1, 1, 4, 7, 10, 10, 15, 20, 25, 30, 42, 49, 56}, 80]
%Y Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309689, A309690, A309692, A309694.
%K nonn,easy
%O 0,8
%A _Wesley Ivan Hurt_, Aug 12 2019
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