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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.

%I A026923 #25 Sep 07 2019 01:38:40
%S A026923 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,
%T A026923 37,35,43,40,48,45,54,51,60,57,67,63,73,70,81,77,88,84,96,92,104,100,
%U A026923 113,108,121,117,131
%N 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.
%H A026923 R. J. Mathar, <a href="/A026923/b026923.txt">Table of n, a(n) for n = 1..1000</a>
%F A026923 a(n) + A026927(n) = A069905(n). - _R. J. Mathar_, Aug 22 2019
%F A026923 Conjectures from _Colin Barker_, Sep 01 2019: (Start)
%F A026923 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)).
%F A026923 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.
%F A026923 (End)
%e A026923 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A026923                                                           1+1+8
%e A026923                                                    1+1+7  1+2+7
%e A026923                                                    1+2+6  1+3+6
%e A026923                                             1+1+6  1+3+5  1+4+5
%e A026923                                      1+1+5  1+2+5  1+4+4  2+2+6
%e A026923                               1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
%e A026923                        1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
%e A026923          1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
%e A026923 -----------------------------------------------------------------------
%e A026923   n  |     3      4      5      6      7      8      9     10      ...
%e A026923 -----------------------------------------------------------------------
%e A026923 a(n) |     1      0      1      1      3      2      4      3      ...
%e A026923 -----------------------------------------------------------------------
%e A026923 - _Wesley Ivan Hurt_, Sep 06 2019
%p A026923 A026923 := proc(n)
%p A026923     local a,p1,p2,p3 ;
%p A026923     a := 0 ;
%p A026923     for p1 from 0 to n do
%p A026923         for p2 from 0 to (n-p1)/2 do
%p A026923             p3 := (n-p1-2*p2)/3 ;
%p A026923             if type(p3,'integer') and p3 >=1 and type(p1+p2+p3,'odd') then
%p A026923                 a := a+1 ;
%p A026923             end if:
%p A026923         end do:
%p A026923     end do:
%p A026923     a;
%p A026923 end proc: # _R. J. Mathar_, Aug 22 2019
%Y A026923 3rd column of A026920.
%Y A026923 Cf. A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.
%K A026923 nonn
%O A026923 1,7
%A A026923 _Clark Kimberling_