A309688 Sum of the odd parts appearing among the second largest parts of the partitions of n into 3 parts.

0, 0, 0, 1, 1, 1, 1, 4, 7, 10, 10, 15, 20, 25, 30, 42, 49, 56, 63, 79, 95, 111, 120, 140, 160, 180, 200, 233, 257, 281, 305, 344, 383, 422, 450, 495, 540, 585, 630, 694, 745, 796, 847, 919, 991, 1063, 1120, 1200, 1280, 1360, 1440, 1545, 1633, 1721, 1809

Offset

0,8

Links

Table of n, a(n) for n=0..54.

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 mod 2).

From Colin Barker, Aug 23 2019: (Start)

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

Mathematica

Table[Sum[Sum[i * Mod[i, 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, 1, 1, 1, 1, 4, 7, 10, 10, 15, 20, 25, 30, 42, 49, 56}, 80]

Crossrefs

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309689, A309690, A309692, A309694.

Sequence in context: A282848 A093465 A123869 * A072125 A370759 A223024

Adjacent sequences: A309685 A309686 A309687 * A309689 A309690 A309691

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 12 2019

Status

approved