A309686 Sum of the even parts appearing among the smallest parts of the partitions of n into 3 parts.

0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 6, 12, 12, 18, 18, 24, 24, 36, 36, 48, 48, 60, 60, 80, 80, 100, 100, 120, 120, 150, 150, 180, 180, 210, 210, 252, 252, 294, 294, 336, 336, 392, 392, 448, 448, 504, 504, 576, 576, 648, 648, 720, 720, 810, 810, 900, 900, 990

Offset

0,7

Links

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

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-1) mod 2).

From Colin Barker, Aug 23 2019: (Start)

G.f.: 2*x^6 / ((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)

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      0      2      2      4      4      6      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[j*Mod[j - 1, 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, 0, 0, 0, 2, 2, 4, 4, 6, 6, 12, 12, 18}, 80]

Crossrefs

Cf. A026923, A026927, A309683, A309684, A309685, A309687, A309688, A309689, A309690, A309692, A309694.

Sequence in context: A357452 A211511 A211513 * A320008 A127370 A352792

Adjacent sequences: A309683 A309684 A309685 * A309687 A309688 A309689

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 12 2019

Status

approved