A309689 Number of even parts appearing among the second largest parts of the partitions of n into 3 parts.

0, 0, 0, 0, 0, 1, 2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 35, 38, 40, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 77, 80, 84, 88, 92, 96, 100, 104, 108, 112, 117, 122, 126, 130, 135, 140, 145, 150, 155, 160, 165, 170

Offset

0,7

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

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

Formula

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

From Colin Barker, Aug 23 2019: (Start)

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

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) for n>9.

(End)

a(n) = (6*n^2+48*cos(n*Pi/3)-36*cos(n*Pi/2)+16*cos(2*n*Pi/3)-3*(-1)^n-25)/144. - Ilya Gutkovskiy, Oct 29 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) |     0      0      1      2      2      2      3      4      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{2, -2, 2, -1, 0, 1, -2, 2, -2, 1}, {0, 0, 0, 0, 0, 1, 2, 2, 2, 3}, 80]

Prog

(PARI) concat([0, 0, 0, 0, 0], Vec(x^5 / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Aug 23 2019

Crossrefs

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309690, A309692, A309694.

Sequence in context: A163801 A323735 A233583 * A029049 A094983 A238218

Adjacent sequences: A309686 A309687 A309688 * A309690 A309691 A309692

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 12 2019

Status

approved