A309716 Sum of the even parts appearing among the third largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 18, 24, 34, 44, 54, 64, 80, 96, 118, 140, 168, 196, 232, 268, 312, 356, 408, 460, 530, 600, 680, 760, 850, 940, 1052, 1164, 1298, 1432, 1578, 1724, 1896, 2068, 2266, 2464, 2688, 2912, 3166, 3420, 3704, 3988, 4302

Offset

0,8

Links

Table of n, a(n) for n=0..53.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,0,-2,4,-6,6,-6,5,-4,0,4,-5,6,-6,6,-4,2,0,-2,2,-2,2,-2,1).

Formula

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

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-7) + 4*a(n-8) - 6*a(n-9) + 6*a(n-10) - 6*a(n-11) + 5*a(n-12) - 4*a(n-13) + 4*a(n-15) - 5*a(n-16) + 6*a(n-17) - 6*a(n-18) + 6*a(n-19) - 4*a(n-20) + 2*a(n-21) - 2*a(n-23) + 2*a(n-24) - 2*a(n-25) + 2*a(n-26) - 2*a(n-27) + a(n-28) for n > 27. - Wesley Ivan Hurt, Sep 04 2019

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) |      4           6           8          10          12        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 04 2019

Mathematica

Table[Sum[Sum[Sum[j * Mod[j - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
LinearRecurrence[{2, -2, 2, -2, 2, 0, -2, 4, -6, 6, -6, 5, -4, 0, 4, -5, 6, -6, 6, -4, 2, 0, -2, 2, -2, 2, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 18, 24, 34, 44, 54, 64, 80, 96, 118, 140, 168, 196, 232, 268, 312}, 60] (* Wesley Ivan Hurt, Sep 04 2019 *)

Crossrefs

Cf. A309711, A309715.

Sequence in context: A162763 A113242 A343716 * A198186 A379294 A264984

Adjacent sequences: A309713 A309714 A309715 * A309717 A309718 A309719

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 13 2019

Status

approved