A309511 Number of odd parts in the partitions of n into 3 parts.

0, 0, 0, 3, 2, 4, 4, 8, 8, 13, 12, 18, 18, 24, 24, 33, 32, 40, 40, 50, 50, 61, 60, 72, 72, 84, 84, 99, 98, 112, 112, 128, 128, 145, 144, 162, 162, 180, 180, 201, 200, 220, 220, 242, 242, 265, 264, 288, 288, 312, 312, 339, 338, 364, 364, 392, 392, 421, 420

Offset

0,4

Links

Table of n, a(n) for n=0..58.

Index entries for sequences related to partitions

Formula

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

Conjectures from Colin Barker, Aug 06 2019: (Start)

G.f.: x^3*(3 - x + 2*x^2 + x^4 + x^5) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).

a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) for n>10.

(End)

Example

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    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     3      2      4      4      8      8     13     12      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Mod[i, 2] + Mod[j, 2] + Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
Table[Count[Flatten[IntegerPartitions[n, {3}]], _?OddQ], {n, 0, 60}] (* Harvey P. Dale, Jan 16 2022 *)

Crossrefs

Sequence in context: A371180 A147604 A095401 * A195472 A370806 A240538

Adjacent sequences: A309508 A309509 A309510 * A309512 A309513 A309514

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 05 2019

Status

approved