A309684 Sum of the odd parts appearing among the smallest parts of the partitions of n into 3 parts.

0, 0, 0, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 15, 15, 24, 24, 33, 33, 42, 42, 58, 58, 74, 74, 90, 90, 115, 115, 140, 140, 165, 165, 201, 201, 237, 237, 273, 273, 322, 322, 371, 371, 420, 420, 484, 484, 548, 548, 612, 612, 693, 693, 774, 774, 855, 855, 955, 955

Offset

0,6

Links

Jinyuan Wang, Table of n, a(n) for n = 0..5000

Index entries for sequences related to partitions

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

Formula

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

From Colin Barker, Aug 22 2019: (Start)

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

a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-6) - 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-12) + a(n-13) + a(n-14) - a(n-15) for n > 14.

(End)

a(n) = (-4*s^3+(2*t-7)*s^2+(4*t-1)*s+2*t+2)/2, where s = floor((n-3)/6) and t = floor((n-3)/2). - Wesley Ivan Hurt, Oct 27 2021

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      7      7      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[j*Mod[j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{1, 1, -1, 0, 0, 2, -2, -2, 2, 0, 0, -1, 1, 1, -1}, {0, 0, 0, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 15, 15}, 20] (* Wesley Ivan Hurt, Aug 29 2019 *)

Prog

(PARI) a(n) = sum(j = 1, floor(n/3), sum(i = j, floor((n-j)/2), j * (j%2))); \\ Jinyuan Wang, Aug 29 2019

Crossrefs

Cf. A026923, A026927, A309683, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

Sequence in context: A295510 A324520 A307737 * A330950 A032060 A241385

Adjacent sequences: A309681 A309682 A309683 * A309685 A309686 A309687

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 12 2019

Status

approved