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A308733
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Sum of the smallest parts of the partitions of n into 4 parts.
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4
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0, 0, 0, 0, 1, 1, 2, 3, 6, 7, 11, 14, 21, 25, 34, 41, 55, 64, 81, 95, 119, 136, 165, 189, 227, 256, 301, 339, 396, 441, 507, 564, 645, 711, 804, 885, 996, 1089, 1215, 1326, 1474, 1600, 1766, 1914, 2106, 2272, 2486, 2678, 2922, 3136, 3406, 3650, 3955, 4225, 4560
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OFFSET
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0,7
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LINKS
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FORMULA
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a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} k.
G.f.: x^4 / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) + a(n-4) - 3*a(n-5) - a(n-6) + a(n-8) + 3*a(n-9) - a(n-10) - a(n-12) - a(n-13) + a(n-14) for n>13.
(End)
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EXAMPLE
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Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
1+1+1+9
1+1+2+8
1+1+3+7
1+1+4+6
1+1+1+8 1+1+5+5
1+1+2+7 1+2+2+7
1+1+1+7 1+1+3+6 1+2+3+6
1+1+2+6 1+1+4+5 1+2+4+5
1+1+3+5 1+2+2+6 1+3+3+5
1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4
1+1+1+5 1+1+2+5 1+2+2+5 1+2+4+4 2+2+2+6
1+1+2+4 1+1+3+4 1+2+3+4 1+3+3+4 2+2+3+5
1+1+3+3 1+2+2+4 1+3+3+3 2+2+2+5 2+2+4+4
1+2+2+3 1+2+3+3 2+2+2+4 2+2+3+4 2+3+3+4
2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3
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n | 8 9 10 11 12 ...
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a(n) | 6 7 11 14 21 ...
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MATHEMATICA
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Table[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {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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