A309519 Sum of the even parts in the partitions of n into 4 parts.

0, 0, 0, 0, 0, 2, 4, 12, 18, 28, 40, 62, 80, 120, 154, 208, 260, 334, 408, 522, 620, 764, 902, 1090, 1270, 1512, 1742, 2042, 2324, 2694, 3050, 3510, 3944, 4476, 4998, 5640, 6252, 7020, 7752, 8634, 9490, 10502, 11494, 12684, 13816, 15160, 16468, 17996, 19484

Offset

0,6

Links

Table of n, a(n) for n=0..48.

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) + j * ((j-1) mod 2) + k * ((k-1) mod 2) + (n-i-j-k) * ((n-i-j-k-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) |     18          28          40          62          80        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 08 2019

Mathematica

Table[Sum[Sum[Sum[(i * Mod[i - 1, 2] + j * Mod[j - 1, 2] + k * Mod[k - 1, 2] + (n - i - j - k) * Mod[n - i - j - k - 1, 2]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]

Prog

(PARI) a(n) = sum(k=1, n\4, sum(j=k, (n-k)\3, sum(i=j, (n-j-k)\2, (i * ((i-1) % 2) + j * ((j-1) % 2) + k * ((k-1) % 2) + (n-i-j-k) * ((n-i-j-k-1) % 2))))); \\ Michel Marcus, Sep 08 2019

Crossrefs

Sequence in context: A064361 A303403 A064407 * A198392 A052289 A309547

Adjacent sequences: A309516 A309517 A309518 * A309520 A309521 A309522

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 05 2019

Extensions

Data corrected by Wesley Ivan Hurt, Sep 08 2019

Status

approved