A309513 Number of even parts in the partitions of n into 3 parts.
0, 0, 0, 0, 1, 2, 5, 4, 7, 8, 12, 12, 18, 18, 24, 24, 31, 32, 41, 40, 49, 50, 60, 60, 72, 72, 84, 84, 97, 98, 113, 112, 127, 128, 144, 144, 162, 162, 180, 180, 199, 200, 221, 220, 241, 242, 264, 264, 288, 288, 312, 312, 337, 338, 365, 364, 391, 392, 420, 420
Offset: 0
Examples
Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
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n | 3 4 5 6 7 8 9 10 ...
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a(n) | 0 1 2 5 4 7 8 12 ...
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Links
- Ray Chandler, Table of n, a(n) for n = 0..1000
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1,1,-1,0,0,-1,1).
Programs
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Mathematica
Table[Sum[Sum[Mod[i - 1, 2] + Mod[j - 1, 2] + Mod[n - i - j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
Formula
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (((i-1) mod 2) + ((j-1) mod 2) + ((n-i-j-1) mod 2)). [Corrected by Georg Fischer, Mar 11 2025]
From Colin Barker, Aug 06 2019: (Start)
G.f.: x^4*(1 + x + 3*x^2 - x^3 + 2*x^4) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) for n>10. (End)