A308265 Sum of the largest parts in the partitions of n into 3 parts.

0, 0, 1, 2, 5, 9, 15, 22, 34, 45, 62, 81, 104, 129, 163, 195, 237, 282, 333, 387, 454, 518, 596, 678, 768, 862, 973, 1080, 1205, 1335, 1475, 1620, 1786, 1947, 2130, 2319, 2520, 2727, 2959, 3185, 3437, 3696, 3969, 4249, 4558, 4860, 5192, 5532, 5888, 6252

Offset

1,4

Links

Table of n, a(n) for n=1..50.

Index entries for sequences related to partitions

Formula

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

Conjectures from Colin Barker, Jul 16 2019: (Start)

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

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 4*a(n-5) - a(n-6) + 2*a(n-7) + 2*a(n-8) - a(n-10) for n>10.

(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) |     1      2      5      9     15     22     34     45      ...
-----------------------------------------------------------------------

Mathematica

Table[Sum[Sum[n - i - k, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Crossrefs

Cf. A307872.

Sequence in context: A320259 A007982 A011904 * A218914 A047809 A014126

Adjacent sequences: A308262 A308263 A308264 * A308266 A308267 A308268

Keyword

nonn

Author

Wesley Ivan Hurt, May 17 2019

Status

approved