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%I #9 Aug 06 2019 11:18:07
%S 0,0,0,3,2,6,6,18,18,37,34,64,64,100,100,161,156,220,220,310,310,417,
%T 410,542,542,686,686,877,868,1064,1064,1304,1304,1571,1560,1866,1866,
%U 2190,2190,2583,2570,2970,2970,3432,3432,3931,3916,4468,4468,5044,5044
%N Sum of the odd parts in 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 mod 2) + j * (j mod 2) + (n-i-j) * ((n-i-j) mod 2)).
%F Conjectures from _Colin Barker_, Aug 06 2019: (Start)
%F G.f.: x^3*(3 - x + 7*x^2 - x^3 + 13*x^4 + x^5 + 18*x^6 + 19*x^8 - x^9 + 9*x^10 + x^11 + 3*x^12 + x^13) / ((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) | 3 2 6 6 18 18 37 34 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[i*Mod[i, 2] + j*Mod[j, 2] + (n - i - j)*Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
%K nonn
%O 0,4
%A _Wesley Ivan Hurt_, Aug 05 2019
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