A309694 Sum of the even parts appearing among the largest parts of the partitions of n into 3 parts.
0, 0, 0, 0, 2, 2, 6, 4, 14, 14, 28, 24, 48, 44, 74, 68, 112, 106, 158, 144, 214, 206, 286, 268, 370, 352, 466, 444, 584, 562, 716, 680, 864, 838, 1038, 996, 1230, 1188, 1440, 1392, 1682, 1634, 1944, 1876, 2228, 2174, 2548, 2472, 2892, 2816, 3260, 3176, 3670
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 2 2 6 4 14 14 28 ...
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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 - 1, 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, 0, 2, 2, 6, 4, 14, 14, 28, 24, 48, 44, 74, 68, 112, 106, 158}, 80] -
PARI
concat([0,0,0,0], Vec(2*x^4*(1 + 3*x^2 - x^3 + 6*x^4 - x^5 + 7*x^6 - x^7 + 6*x^8 - x^9 + 3*x^10 - x^11 + 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-1) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: 2*x^4*(1 + 3*x^2 - x^3 + 6*x^4 - x^5 + 7*x^6 - x^7 + 6*x^8 - x^9 + 3*x^10 - x^11 + 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)