cp's OEIS Frontend

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

%I A309685 #18 Sep 01 2019 09:22:45
%S A309685 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,
%T A309685 30,35,35,40,40,45,45,51,51,57,57,63,63,70,70,77,77,84,84,92,92,100,
%U A309685 100,108,108,117,117,126,126,135,135,145,145,155,155,165
%N A309685 Number of even parts appearing among the smallest parts of the partitions of n into 3 parts.
%H A309685 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A309685 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1, 0, 0, 1, -1, -1, 1).
%F A309685 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ((j-1) mod 2).
%F A309685 From _Colin Barker_, Aug 23 2019: (Start)
%F A309685 G.f.: x^6 / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)).
%F A309685 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.
%F A309685 (End)
%F A309685 a(n) = A001840(floor((n-4)/2)) for n>=2. - _Joerg Arndt_, Aug 23 2019
%e A309685 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A309685                                                           1+1+8
%e A309685                                                    1+1+7  1+2+7
%e A309685                                                    1+2+6  1+3+6
%e A309685                                             1+1+6  1+3+5  1+4+5
%e A309685                                      1+1+5  1+2+5  1+4+4  2+2+6
%e A309685                               1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
%e A309685                        1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
%e A309685          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 A309685 -----------------------------------------------------------------------
%e A309685   n  |     3      4      5      6      7      8      9     10      ...
%e A309685 -----------------------------------------------------------------------
%e A309685 a(n) |     0      0      0      1      1      2      2      3      ...
%e A309685 -----------------------------------------------------------------------
%t A309685 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 *)
%Y A309685 Cf. A001840, A026923, A026927, A309683, A309684, A309686, A309687, A309688, A309689, A309690, A309692, A309694.
%K A309685 nonn,easy
%O A309685 0,9
%A A309685 _Wesley Ivan Hurt_, Aug 12 2019