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A309796
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Sum of the even parts appearing among the second largest parts of the partitions of n into 4 parts.
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2
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0, 0, 0, 0, 0, 0, 2, 4, 6, 6, 10, 14, 22, 30, 44, 54, 70, 82, 108, 134, 168, 196, 242, 282, 340, 392, 466, 532, 620, 700, 816, 924, 1058, 1174, 1332, 1480, 1670, 1850, 2078, 2284, 2538, 2770, 3070, 3358, 3700, 4016, 4408, 4774, 5216, 5632, 6132, 6602, 7156
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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)} i * ((i-1) mod 2).
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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 6 10 14 22 ...
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MATHEMATICA
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Table[Sum[Sum[Sum[ i*Mod[i - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n, {4}][[;; , 2]], EvenQ]], {n, 0, 60}] (* Harvey P. Dale, Mar 05 2023 *)
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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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