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A309516
Number of odd parts in the partitions of n into 4 parts.
2
0, 0, 0, 0, 4, 3, 6, 7, 12, 14, 22, 25, 36, 40, 52, 59, 76, 85, 104, 116, 140, 154, 182, 200, 232, 254, 290, 316, 360, 389, 436, 471, 524, 564, 624, 669, 736, 786, 858, 915, 996, 1059, 1146, 1216, 1312, 1388, 1492, 1576, 1688, 1780, 1900, 2000, 2132, 2239
OFFSET
0,5
FORMULA
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} ((i mod 2) + (j mod 2) + (k mod 2) + ((n-i-j-k) mod 2)).
Conjectures from Colin Barker, Aug 06 2019: (Start)
G.f.: x^4*(4 - 5*x + 4*x^2 - 2*x^3 + 4*x^4 - 3*x^5 + 2*x^6) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x^4)).
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) - a(n-14) + 2*a(n-15) - a(n-16) for n>15.
(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
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n | 8 9 10 11 12 ...
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a(n) | 12 14 22 25 36 ...
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- Wesley Ivan Hurt, Sep 07 2019
MATHEMATICA
Table[Sum[Sum[Sum[(Mod[i, 2] + Mod[j, 2] + Mod[k, 2] + Mod[n - i - j - k, 2]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]
CROSSREFS
Sequence in context: A266576 A103476 A021700 * A211362 A211246 A197829
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Aug 05 2019
STATUS
approved