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A309515 Sum of the even parts of the partitions of n into 3 parts. 0
0, 0, 0, 0, 2, 4, 12, 10, 22, 26, 46, 46, 80, 82, 124, 124, 180, 188, 266, 260, 350, 360, 470, 470, 610, 614, 770, 770, 952, 966, 1186, 1176, 1416, 1432, 1704, 1704, 2022, 2028, 2370, 2370, 2750, 2770, 3204, 3190, 3652, 3674, 4180, 4180, 4748, 4756, 5356 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..50.

Index entries for sequences related to partitions

FORMULA

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

Conjectures from Colin Barker, Aug 06 2019: (Start)

G.f.: 2*x^4*(1 + x + 5*x^2 + 9*x^4 + 9*x^6 + 7*x^8 + 4*x^10 - x^11 + x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).

a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + 3*a(n-6) - 3*a(n-7) + 2*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14) + a(n-15) + a(n-16) - a(n-17) + a(n-18) - a(n-19) for  n>18.

(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      2      4     12     10     22     26     46      ...

-----------------------------------------------------------------------

MATHEMATICA

Table[Sum[Sum[i * Mod[i - 1, 2] + j * Mod[j - 1, 2] + (n - i - j) * Mod[n - i - j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]]

CROSSREFS

Sequence in context: A181817 A000348 A168342 * A141668 A087796 A268494

Adjacent sequences:  A309512 A309513 A309514 * A309516 A309517 A309518

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Aug 05 2019

STATUS

approved

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Last modified February 25 22:37 EST 2021. Contains 341618 sequences. (Running on oeis4.)