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A309512
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Sum of the odd parts in the partitions of n into 3 parts.
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0
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0, 0, 0, 3, 2, 6, 6, 18, 18, 37, 34, 64, 64, 100, 100, 161, 156, 220, 220, 310, 310, 417, 410, 542, 542, 686, 686, 877, 868, 1064, 1064, 1304, 1304, 1571, 1560, 1866, 1866, 2190, 2190, 2583, 2570, 2970, 2970, 3432, 3432, 3931, 3916, 4468, 4468, 5044, 5044
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OFFSET
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0,4
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LINKS
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FORMULA
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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)).
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).
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.
(End)
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EXAMPLE
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Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
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n | 3 4 5 6 7 8 9 10 ...
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a(n) | 3 2 6 6 18 18 37 34 ...
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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