A309690 Sum of the even parts appearing among the second largest parts of the partitions of n into 3 parts.

0, 0, 0, 0, 0, 2, 4, 4, 4, 8, 12, 16, 20, 26, 32, 38, 44, 58, 72, 80, 88, 106, 124, 142, 160, 182, 204, 226, 248, 284, 320, 346, 372, 414, 456, 498, 540, 588, 636, 684, 732, 800, 868, 922, 976, 1052, 1128, 1204, 1280, 1364, 1448, 1532, 1616, 1726, 1836, 1928

Offset

0,6

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,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).

Formula

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

From Colin Barker, Aug 23 2019: (Start)

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

a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 3*a(n-4) + 2*a(n-5) + a(n-6) - 4*a(n-7) + 6*a(n-8) - 8*a(n-9) + 6*a(n-10) - 4*a(n-11) + a(n-12) + 2*a(n-13) - 3*a(n-14) + 4*a(n-15) - 3*a(n-16) + 2*a(n-17) - a(n-18) for n>17.

(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) |     0      0      2      4      4      4      8     12      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[i * Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{2, -3, 4, -3, 2, 1, -4, 6, -8, 6, -4, 1, 2, -3, 4, -3, 2, -1}, {0, 0, 0, 0, 0, 2, 4, 4, 4, 8, 12, 16, 20, 26, 32, 38, 44, 58}, 80]

Prog

(PARI) concat([0, 0, 0, 0, 0], Vec(2*x^5*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^60))) \\ Colin Barker, Aug 23 2019

Crossrefs

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309692, A309694.

Sequence in context: A342272 A182635 A188346 * A173531 A122788 A078003

Adjacent sequences: A309687 A309688 A309689 * A309691 A309692 A309693

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 12 2019

Status

approved