A309796 Sum of the even parts appearing among the second largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 0, 0, 2, 4, 6, 6, 10, 14, 22, 30, 44, 54, 70, 82, 108, 134, 168, 196, 242, 282, 340, 392, 466, 532, 620, 700, 816, 924, 1058, 1174, 1332, 1480, 1670, 1850, 2078, 2284, 2538, 2770, 3070, 3358, 3700, 4016, 4408, 4774, 5216, 5632, 6132, 6602, 7156

Offset

0,7

Links

Table of n, a(n) for n=0..52.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} i * ((i-1) mod 2).

Example

Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |      6           6          10          14          22        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 11 2019

Mathematica

Table[Sum[Sum[Sum[ i*Mod[i - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n, {4}][[;; , 2]], EvenQ]], {n, 0, 60}] (* Harvey P. Dale, Mar 05 2023 *)

Crossrefs

Sequence in context: A254732 A299541 A066820 * A359671 A222733 A364828

Adjacent sequences: A309793 A309794 A309795 * A309797 A309798 A309799

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 17 2019

Status

approved