A309834 Sum of the even parts appearing among the smallest parts of the partitions of n into 5 parts.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 10, 12, 18, 22, 30, 36, 50, 58, 76, 90, 114, 132, 164, 188, 228, 260, 314, 354, 420, 474, 556, 622, 722, 804, 924, 1024, 1172, 1292, 1466, 1614, 1820, 1994, 2236, 2442, 2722, 2964, 3294, 3574, 3952, 4282, 4716, 5094
Offset: 0
Examples
Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
1+1+1+1+10
1+1+1+2+9
1+1+1+3+8
1+1+1+4+7
1+1+1+5+6
1+1+1+1+9 1+1+2+2+8
1+1+1+2+8 1+1+2+3+7
1+1+1+3+7 1+1+2+4+6
1+1+1+4+6 1+1+2+5+5
1+1+1+5+5 1+1+3+3+6
1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
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n | 10 11 12 13 14 ...
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a(n) | 2 2 4 6 10 ...
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Links
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,1,-2,-2,0,0,4,0,0,-2,-2,1,1,1,0,0,-2,0,0,1,1,-1).
Programs
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Mathematica
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, 1, -2, -2, 0, 0, 4, 0, 0, -2, -2, 1, 1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 10, 12, 18, 22, 30, 36, 50, 58, 76, 90, 114, 132, 164, 188, 228, 260}, 50]
Formula
a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} l * ((l-1) mod 2).
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) + a(n-10) - 2*a(n-11) - 2*a(n-12) + 4*a(n-15) - 2*a(n-18) - 2*a(n-19) + a(n-20) + a(n-21) + a(n-22) - 2*a(n-25) + a(n-28) + a(n-29) - a(n-30) for n > 29.