A309794 Sum of the odd parts appearing among the second largest parts of the partitions of n into 4 parts.

0, 0, 0, 0, 1, 1, 1, 1, 4, 7, 13, 16, 24, 29, 39, 49, 71, 88, 112, 131, 166, 196, 242, 281, 340, 392, 464, 529, 627, 709, 815, 912, 1048, 1175, 1341, 1487, 1678, 1849, 2065, 2270, 2539, 2784, 3080, 3352, 3696, 4016, 4408, 4772, 5216, 5632, 6128, 6596, 7169

Offset

0,9

Links

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

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)} i * (i 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.

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           7          13          16          24        ...
--------------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Sum[i* Mod[i, 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, 1, 1, 1, 1, 4, 7, 13, 16, 24, 29, 39, 49, 71, 88, 112, 131, 166, 196, 242, 281, 340, 392, 464, 529}, 60]

Crossrefs

Cf. A309711, A309715.

Sequence in context: A310815 A310816 A310817 * A048297 A061201 A219907

Adjacent sequences: A309791 A309792 A309793 * A309795 A309796 A309797

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 17 2019

Status

approved