A026927 -id:A026927 - cp's OEIS frontend

A026923 Number of partitions of n into an odd number of parts, the greatest being 3; also, a(n+5) = number of partitions of n+2 into an even number of parts, each <= 3.

0, 0, 1, 0, 1, 1, 3, 2, 4, 3, 6, 5, 8, 7, 11, 9, 13, 12, 17, 15, 20, 18, 24, 22, 28, 26, 33, 30, 37, 35, 43, 40, 48, 45, 54, 51, 60, 57, 67, 63, 73, 70, 81, 77, 88, 84, 96, 92, 104, 100, 113, 108, 121, 117, 131

Offset: 1

Clark Kimberling

nonn

			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      0      1      1      3      2      4      3      ...
-----------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 06 2019

R. J. Mathar, Table of n, a(n) for n = 1..1000

3rd column of A026920.

Cf. A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

A026923 := proc(n)
    local a,p1,p2,p3 ;
    a := 0 ;
    for p1 from 0 to n do
        for p2 from 0 to (n-p1)/2 do
            p3 := (n-p1-2*p2)/3 ;
            if type(p3,'integer') and p3 >=1 and type(p1+p2+p3,'odd') then
                a := a+1 ;
            end if:
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Aug 22 2019

a(n) + A026927(n) = A069905(n). - R. J. Mathar, Aug 22 2019
Conjectures from Colin Barker, Sep 01 2019: (Start)
G.f.: x^3*(1 - x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) for n>11.
(End)

A309684 Sum of the odd parts appearing among the smallest parts of the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 15, 15, 24, 24, 33, 33, 42, 42, 58, 58, 74, 74, 90, 90, 115, 115, 140, 140, 165, 165, 201, 201, 237, 237, 273, 273, 322, 322, 371, 371, 420, 420, 484, 484, 548, 548, 612, 612, 693, 693, 774, 774, 855, 855, 955, 955

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

			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      2      2      3      3      7      7      ...
-----------------------------------------------------------------------

Jinyuan Wang, Table of n, a(n) for n = 0..5000
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).

Cf. A026923, A026927, A309683, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

Table[Sum[Sum[j*Mod[j, 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, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 15, 15}, 20] (* Wesley Ivan Hurt, Aug 29 2019 *)

a(n) = sum(j = 1, floor(n/3), sum(i = j, floor((n-j)/2), j * (j%2))); \\ Jinyuan Wang, Aug 29 2019

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} j * (j mod 2).
From Colin Barker, Aug 22 2019: (Start)
G.f.: x^3*(1 + x^2)*(1 - x^2 + x^4) / ((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)
a(n) = (-4*s^3+(2*t-7)*s^2+(4*t-1)*s+2*t+2)/2, where s = floor((n-3)/6) and t = floor((n-3)/2). - Wesley Ivan Hurt, Oct 27 2021

A309689 Number of even parts appearing among the second largest parts of the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 35, 38, 40, 42, 45, 48, 51, 54, 57, 60, 63, 66, 70, 74, 77, 80, 84, 88, 92, 96, 100, 104, 108, 112, 117, 122, 126, 130, 135, 140, 145, 150, 155, 160, 165, 170

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

			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      1      2      2      2      3      4      ...
-----------------------------------------------------------------------

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,-2,2,-1,0,1,-2,2,-2,1).

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309690, A309692, A309694.

Table[Sum[Sum[Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{2, -2, 2, -1, 0, 1, -2, 2, -2, 1}, {0, 0, 0, 0, 0, 1, 2, 2, 2, 3}, 80]

concat([0,0,0,0,0], Vec(x^5 / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Aug 23 2019

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ((i-1) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^5 / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) for n>9.
(End)
a(n) = (6*n^2+48*cos(n*Pi/3)-36*cos(n*Pi/2)+16*cos(2*n*Pi/3)-3*(-1)^n-25)/144. - Ilya Gutkovskiy, Oct 29 2021

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

Original entry on oeis.org

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

Wesley Ivan Hurt, Aug 12 2019

			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      ...
-----------------------------------------------------------------------

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).

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

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]

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

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)

A309692 Sum of the odd parts appearing among the largest parts of the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 3, 11, 8, 20, 17, 38, 33, 60, 55, 95, 83, 131, 124, 189, 173, 248, 232, 328, 308, 416, 396, 529, 496, 643, 619, 795, 756, 948, 909, 1134, 1089, 1332, 1287, 1567, 1503, 1803, 1752, 2093, 2021, 2384, 2312, 2720, 2640, 3072, 2992, 3473, 3368

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

			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      0      3      3     11      8     20     17      ...
-----------------------------------------------------------------------

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 (1,-1,1,1,-1,3,-3,2,-2,-2,2,-3,3,-1,1,1,-1,1,-1).

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309694.

Table[Sum[Sum[ (n - i - j) * Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{1, -1, 1, 1, -1, 3, -3, 2, -2, -2, 2, -3, 3, -1, 1, 1, -1, 1, -1}, {0, 0, 0, 1, 0, 3, 3, 11, 8, 20, 17, 38, 33, 60, 55, 95, 83, 131, 124}, 80]
Table[Total[Select[IntegerPartitions[n,{3}][[;;,1]],OddQ]],{n,0,60}] (* Harvey P. Dale, Oct 13 2023 *)

concat([0,0,0], Vec(x^3*(1 - x + 4*x^2 - x^3 + 10*x^4 - 2*x^5 + 14*x^6 - 3*x^7 + 14*x^8 - 3*x^9 + 8*x^10 + 3*x^12) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^40))) \\ Colin Barker, Aug 23 2019

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (n-i-j) * ((n-i-j) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^3*(1 - x + 4*x^2 - x^3 + 10*x^4 - 2*x^5 + 14*x^6 - 3*x^7 + 14*x^8 - 3*x^9 + 8*x^10 + 3*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)

A309694 Sum of the even parts appearing among the largest parts of the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 6, 4, 14, 14, 28, 24, 48, 44, 74, 68, 112, 106, 158, 144, 214, 206, 286, 268, 370, 352, 466, 444, 584, 562, 716, 680, 864, 838, 1038, 996, 1230, 1188, 1440, 1392, 1682, 1634, 1944, 1876, 2228, 2174, 2548, 2472, 2892, 2816, 3260, 3176, 3670

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

			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      2      6      4     14     14     28      ...
-----------------------------------------------------------------------

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 (1,-1,1,1,-1,3,-3,2,-2,-2,2,-3,3,-1,1,1,-1,1,-1).

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692.

Table[Sum[Sum[(n - i - j) * Mod[n - i - j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{1, -1, 1, 1, -1, 3, -3, 2, -2, -2, 2, -3, 3, -1, 1, 1, -1, 1, -1}, {0, 0, 0, 0, 2, 2, 6, 4, 14, 14, 28, 24, 48, 44, 74, 68, 112, 106, 158}, 80]

concat([0,0,0,0], Vec(2*x^4*(1 + 3*x^2 - x^3 + 6*x^4 - x^5 + 7*x^6 - x^7 + 6*x^8 - x^9 + 3*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) + O(x^40))) \\ Colin Barker, Aug 23 2019

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (n-i-j) * ((n-i-j-1) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: 2*x^4*(1 + 3*x^2 - x^3 + 6*x^4 - x^5 + 7*x^6 - x^7 + 6*x^8 - x^9 + 3*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)

A309683 Number of odd parts appearing among the smallest parts of the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 12, 12, 15, 15, 18, 18, 22, 22, 26, 26, 30, 30, 35, 35, 40, 40, 45, 45, 51, 51, 57, 57, 63, 63, 70, 70, 77, 77, 84, 84, 92, 92, 100, 100, 108, 108, 117, 117, 126, 126, 135, 135, 145, 145, 155, 155, 165, 165, 176

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

			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      2      2      3      3      5      5      ...
-----------------------------------------------------------------------

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,1,-1,-1,1).

Cf. A026923, A026927, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

Table[Sum[Sum[Mod[j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 0, 0, 1, 1, 2, 2, 3, 3}, 50] (* Wesley Ivan Hurt, Aug 28 2019 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (j mod 2).
From Colin Barker, Aug 22 2019: (Start)
G.f.: x^3 / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) for n>8.
(End)

A309685 Number of even parts appearing among the smallest parts of the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 12, 12, 15, 15, 18, 18, 22, 22, 26, 26, 30, 30, 35, 35, 40, 40, 45, 45, 51, 51, 57, 57, 63, 63, 70, 70, 77, 77, 84, 84, 92, 92, 100, 100, 108, 108, 117, 117, 126, 126, 135, 135, 145, 145, 155, 155, 165

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

			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      1      1      2      2      3      ...
-----------------------------------------------------------------------

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 0, 0, 1, -1, -1, 1).

Cf. A001840, A026923, A026927, A309683, A309684, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 0, 0, 0, 0, 0, 1, 1, 2}, 80] (* Wesley Ivan Hurt, Aug 30 2019 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ((j-1) mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^6 / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9) for n>8.
(End)
a(n) = A001840(floor((n-4)/2)) for n>=2. - Joerg Arndt, Aug 23 2019

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

Original entry on oeis.org

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

Wesley Ivan Hurt, Aug 12 2019

			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      ...
-----------------------------------------------------------------------

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).

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

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]

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)

A309687 Number of odd parts appearing among the second largest parts of the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 40, 43, 46, 48, 51, 54, 57, 60, 64, 67, 70, 73, 77, 81, 85, 88, 92, 96, 100, 104, 109, 113, 117, 121, 126, 131, 136, 140, 145, 150, 155, 160, 166

Offset: 0

Wesley Ivan Hurt, Aug 12 2019

			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      2      3      4      4      ...
-----------------------------------------------------------------------

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1,0,1,-2,2,-2,1).

Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309688, A309689, A309690, A309692, A309694.

Table[Sum[Sum[Mod[i, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
LinearRecurrence[{2, -2, 2, -1, 0, 1, -2, 2, -2, 1}, {0, 0, 0, 1, 1, 1, 1, 2, 3, 4}, 80]

a(n) = sum(j=1, n\3, sum(i=j, (n-j)\2, i % 2)); \\ Michel Marcus, Aug 23 2019

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (i mod 2).
From Colin Barker, Aug 23 2019: (Start)
G.f.: x^3*(1 - x + x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) for n>9.
(End)

cp's OEIS Frontend

A026923 Number of partitions of n into an odd number of parts, the greatest being 3; also, a(n+5) = number of partitions of n+2 into an even number of parts, each <= 3.

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A309684 Sum of the odd parts appearing among the smallest parts of the partitions of n into 3 parts.

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A309689 Number of even parts appearing among the second largest parts of the partitions of n into 3 parts.

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

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A309692 Sum of the odd parts appearing among the largest parts of the partitions of n into 3 parts.

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A309694 Sum of the even parts appearing among the largest parts of the partitions of n into 3 parts.

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A309683 Number of odd parts appearing among the smallest parts of the partitions of n into 3 parts.

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A309685 Number of even parts appearing among the smallest parts of the partitions of n into 3 parts.

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