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A309711
Sum of the even parts appearing among the smallest parts of the partitions of n into 4 parts.
6
0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 8, 10, 14, 16, 24, 28, 36, 44, 54, 62, 76, 86, 106, 120, 142, 162, 188, 210, 242, 268, 310, 342, 388, 430, 482, 528, 590, 642, 718, 780, 862, 938, 1030, 1112, 1218, 1310, 1434, 1540, 1674, 1798, 1946, 2080, 2246, 2394
OFFSET
0,9
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} k * ((k-1) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: 2*x^8 / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)^2).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-12) + 2*a(n-13) - 2*a(n-14) - a(n-16) + a(n-17) + a(n-18) - a(n-20) - a(n-21) + a(n-22) for n>21.
(End)
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
--------------------------------------------------------------------------
n | 8 9 10 11 12 ...
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a(n) | 2 2 4 6 8 ...
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MATHEMATICA
Table[Sum[Sum[Sum[k * Mod[k - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
LinearRecurrence[{1, 1, 0, -1, -1, 1, 0, 2, -2, -2, 0, 2, 2, -2, 0, -1, 1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 8, 10, 14, 16, 24, 28, 36, 44, 54, 62}, 50]
PROG
(PARI) concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(2*x^8 / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)*(1 + x^4)^2) + O(x^60))) \\ Colin Barker, Aug 23 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 13 2019
STATUS
approved