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A309831
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Number of even parts appearing among the smallest parts of the partitions of n into 5 parts.
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3
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 24, 28, 36, 42, 52, 60, 73, 83, 99, 112, 132, 148, 172, 192, 221, 245, 279, 308, 348, 382, 429, 469, 523, 570, 632, 686, 757, 819, 899, 970, 1061, 1141, 1243, 1334, 1448, 1550, 1677, 1791, 1932
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OFFSET
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0,13
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,0,-1,-1,0,0,2,0,0,-1,-1,1).
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FORMULA
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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-1) mod 2).
G.f.: x^10 / ((1 - x)^5*(1 + x)^3*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-11) - a(n-12) + 2*a(n-15) - a(n-18) - a(n-19) + a(n-20) for n>19.
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EXAMPLE
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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
--------------------------------------------------------------------------
n | 10 11 12 13 14 ...
--------------------------------------------------------------------------
a(n) | 1 1 2 3 5 ...
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MATHEMATICA
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LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, 0, -1, -1, 0, 0, 2, 0,
0, -1, -1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9,
11, 15, 18}, 50]
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PROG
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(PARI) concat([0, 0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(x^10 / ((1 - x)^5*(1 + x)^3*(1 + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^80))) \\ Colin Barker, Oct 10 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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