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

Author

Clark Kimberling

Keywords

• nonn

Examples

			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
		

Links

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

Crossrefs

3rd column of A026920.

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

Programs

Maple

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

Formula

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)