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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.
13
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
1,7
LINKS
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)
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 0 1 1 3 2 4 3 ...
-----------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 06 2019
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
KEYWORD
nonn
STATUS
approved