A308266 Sum of the middle parts in the partitions of n into 3 parts.

0, 0, 1, 1, 3, 5, 8, 11, 18, 22, 31, 40, 51, 62, 80, 93, 114, 135, 159, 183, 217, 244, 282, 320, 362, 404, 459, 505, 565, 625, 690, 755, 836, 906, 993, 1080, 1173, 1266, 1378, 1477, 1596, 1715, 1841, 1967, 2115, 2248, 2404, 2560, 2724, 2888, 3077, 3249, 3447

Offset: 1

Author

Wesley Ivan Hurt, May 17 2019

Keywords

• nonn

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      3      5      8     11     18     22      ...
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Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A308265.

Programs

Mathematica

Table[Sum[Sum[i, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
Table[Total[IntegerPartitions[n,{3}][[All,2]]],{n,60}] (* Harvey P. Dale, Jul 09 2020 *)

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} i.

Conjectures from Colin Barker, Jul 16 2019: (Start)

G.f.: x^3*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 + x + x^2)^2).

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) - a(n-6) + 2*a(n-7) + 2*a(n-8) - a(n-10) for n>10.

(End)