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%I #20 Sep 01 2019 09:23:00
%S 0,0,0,0,0,0,2,2,4,4,6,6,12,12,18,18,24,24,36,36,48,48,60,60,80,80,
%T 100,100,120,120,150,150,180,180,210,210,252,252,294,294,336,336,392,
%U 392,448,448,504,504,576,576,648,648,720,720,810,810,900,900,990
%N Sum of the even 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_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-1) mod 2).
%F From _Colin Barker_, Aug 23 2019: (Start)
%F G.f.: 2*x^6 / ((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)
%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) | 0 0 0 2 2 4 4 6 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[j*Mod[j - 1, 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, 0, 0, 0, 2, 2, 4, 4, 6, 6, 12, 12, 18}, 80]
%Y Cf. A026923, A026927, A309683, A309684, A309685, A309687, A309688, A309689, A309690, A309692, A309694.
%K nonn,easy
%O 0,7
%A _Wesley Ivan Hurt_, Aug 12 2019
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