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%I #7 Aug 06 2019 07:33:58
%S 0,0,0,0,2,4,12,10,22,26,46,46,80,82,124,124,180,188,266,260,350,360,
%T 470,470,610,614,770,770,952,966,1186,1176,1416,1432,1704,1704,2022,
%U 2028,2370,2370,2750,2770,3204,3190,3652,3674,4180,4180,4748,4756,5356
%N Sum of the even parts of 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 * ((i-1) mod 2) + j * ((j-1) mod 2) + (n-i-j) * ((n-i-j-1) mod 2)).
%F Conjectures from _Colin Barker_, Aug 06 2019: (Start)
%F G.f.: 2*x^4*(1 + x + 5*x^2 + 9*x^4 + 9*x^6 + 7*x^8 + 4*x^10 - x^11 + x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).
%F a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + 3*a(n-6) - 3*a(n-7) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14) + a(n-15) + a(n-16) - a(n-17) + a(n-18) - a(n-19) for n>18.
%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 2 4 12 10 22 26 46 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[i * Mod[i - 1, 2] + j * Mod[j - 1, 2] + (n - i - j) * Mod[n - i - j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]]
%K nonn
%O 0,5
%A _Wesley Ivan Hurt_, Aug 05 2019
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