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A309716
Sum of the even parts appearing among the third largest parts of the partitions of n into 4 parts.
0
0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 18, 24, 34, 44, 54, 64, 80, 96, 118, 140, 168, 196, 232, 268, 312, 356, 408, 460, 530, 600, 680, 760, 850, 940, 1052, 1164, 1298, 1432, 1578, 1724, 1896, 2068, 2266, 2464, 2688, 2912, 3166, 3420, 3704, 3988, 4302
OFFSET
0,8
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} j * ((j-1) mod 2).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-7) + 4*a(n-8) - 6*a(n-9) + 6*a(n-10) - 6*a(n-11) + 5*a(n-12) - 4*a(n-13) + 4*a(n-15) - 5*a(n-16) + 6*a(n-17) - 6*a(n-18) + 6*a(n-19) - 4*a(n-20) + 2*a(n-21) - 2*a(n-23) + 2*a(n-24) - 2*a(n-25) + 2*a(n-26) - 2*a(n-27) + a(n-28) for n > 27. - Wesley Ivan Hurt, Sep 04 2019
EXAMPLE
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) | 4 6 8 10 12 ...
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- Wesley Ivan Hurt, Sep 04 2019
MATHEMATICA
Table[Sum[Sum[Sum[j * Mod[j - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
LinearRecurrence[{2, -2, 2, -2, 2, 0, -2, 4, -6, 6, -6, 5, -4, 0, 4, -5, 6, -6, 6, -4, 2, 0, -2, 2, -2, 2, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 18, 24, 34, 44, 54, 64, 80, 96, 118, 140, 168, 196, 232, 268, 312}, 60] (* Wesley Ivan Hurt, Sep 04 2019 *)
CROSSREFS
Sequence in context: A162763 A113242 A343716 * A198186 A264984 A194390
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Aug 13 2019
STATUS
approved