A309692 Sum of the odd parts appearing among the largest parts of the partitions of n into 3 parts.
0, 0, 0, 1, 0, 3, 3, 11, 8, 20, 17, 38, 33, 60, 55, 95, 83, 131, 124, 189, 173, 248, 232, 328, 308, 416, 396, 529, 496, 643, 619, 795, 756, 948, 909, 1134, 1089, 1332, 1287, 1567, 1503, 1803, 1752, 2093, 2021, 2384, 2312, 2720, 2640, 3072, 2992, 3473, 3368
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) | 1 0 3 3 11 8 20 17 ...
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Links
- Colin Barker, 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,-1,1,1,-1,3,-3,2,-2,-2,2,-3,3,-1,1,1,-1,1,-1).
Crossrefs
Programs
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Mathematica
Table[Sum[Sum[ (n - i - j) * Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}] LinearRecurrence[{1, -1, 1, 1, -1, 3, -3, 2, -2, -2, 2, -3, 3, -1, 1, 1, -1, 1, -1}, {0, 0, 0, 1, 0, 3, 3, 11, 8, 20, 17, 38, 33, 60, 55, 95, 83, 131, 124}, 80] Table[Total[Select[IntegerPartitions[n,{3}][[;;,1]],OddQ]],{n,0,60}] (* Harvey P. Dale, Oct 13 2023 *) -
PARI
concat([0,0,0], Vec(x^3*(1 - x + 4*x^2 - x^3 + 10*x^4 - 2*x^5 + 14*x^6 - 3*x^7 + 14*x^8 - 3*x^9 + 8*x^10 + 3*x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^40))) \\ Colin Barker, Aug 23 2019
Formula
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (n-i-j) * ((n-i-j) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^3*(1 - x + 4*x^2 - x^3 + 10*x^4 - 2*x^5 + 14*x^6 - 3*x^7 + 14*x^8 - 3*x^9 + 8*x^10 + 3*x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + 3*a(n-6) - 3*a(n-7) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14) + a(n-15) + a(n-16) - a(n-17) + a(n-18) - a(n-19) for n>18.
(End)