A309683 Number of odd parts appearing among the smallest parts of the partitions of n into 3 parts.

0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 12, 12, 15, 15, 18, 18, 22, 22, 26, 26, 30, 30, 35, 35, 40, 40, 45, 45, 51, 51, 57, 57, 63, 63, 70, 70, 77, 77, 84, 84, 92, 92, 100, 100, 108, 108, 117, 117, 126, 126, 135, 135, 145, 145, 155, 155, 165, 165, 176

Offset

0,6

Links

Table of n, a(n) for n=0..63.

Index entries for sequences related to partitions

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

Formula

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

From Colin Barker, Aug 22 2019: (Start)

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

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) for n>8.

(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) |     1      1      2      2      3      3      5      5      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Mod[j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 0, 0, 1, 1, 2, 2, 3, 3}, 50] (* Wesley Ivan Hurt, Aug 28 2019 *)

Crossrefs

Cf. A026923, A026927, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

Sequence in context: A121261 A085885 A309685 * A283529 A064986 A349675

Adjacent sequences: A309680 A309681 A309682 * A309684 A309685 A309686

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 12 2019

Status

approved