A309688 Sum of the odd parts appearing among the second largest parts of the partitions of n into 3 parts.
0, 0, 0, 1, 1, 1, 1, 4, 7, 10, 10, 15, 20, 25, 30, 42, 49, 56, 63, 79, 95, 111, 120, 140, 160, 180, 200, 233, 257, 281, 305, 344, 383, 422, 450, 495, 540, 585, 630, 694, 745, 796, 847, 919, 991, 1063, 1120, 1200, 1280, 1360, 1440, 1545, 1633, 1721, 1809
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 1 1 1 4 7 10 10 ...
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Links
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).
Crossrefs
Programs
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Mathematica
Table[Sum[Sum[i * Mod[i, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}] LinearRecurrence[{2, -3, 4, -3, 2, 1, -4, 6, -8, 6, -4, 1, 2, -3, 4, -3, 2, -1}, {0, 0, 0, 1, 1, 1, 1, 4, 7, 10, 10, 15, 20, 25, 30, 42, 49, 56}, 80]
Formula
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} i * (i mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^3*(1 + x + x^2 + x^3 + x^4)*(1 - 2*x + 3*x^2 - 4*x^3 + 6*x^4 - 4*x^5 + 3*x^6 - 2*x^7 + x^8) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 3*a(n-4) + 2*a(n-5) + a(n-6) - 4*a(n-7) + 6*a(n-8) - 8*a(n-9) + 6*a(n-10) - 4*a(n-11) + a(n-12) + 2*a(n-13) - 3*a(n-14) + 4*a(n-15) - 3*a(n-16) + 2*a(n-17) - a(n-18) for n>17.
(End)