A309689 Number of even parts appearing among the second largest parts of the partitions of n into 3 parts.
0, 0, 0, 0, 0, 1, 2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 35, 38, 40, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 77, 80, 84, 88, 92, 96, 100, 104, 108, 112, 117, 122, 126, 130, 135, 140, 145, 150, 155, 160, 165, 170
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 0 1 2 2 2 3 4 ...
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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 (2,-2,2,-1,0,1,-2,2,-2,1).
Crossrefs
Programs
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Mathematica
Table[Sum[Sum[Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}] LinearRecurrence[{2, -2, 2, -1, 0, 1, -2, 2, -2, 1}, {0, 0, 0, 0, 0, 1, 2, 2, 2, 3}, 80] -
PARI
concat([0,0,0,0,0], Vec(x^5 / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^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)} ((i-1) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^5 / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) for n>9.
(End)
a(n) = (6*n^2+48*cos(n*Pi/3)-36*cos(n*Pi/2)+16*cos(2*n*Pi/3)-3*(-1)^n-25)/144. - Ilya Gutkovskiy, Oct 29 2021