cp's OEIS Frontend

A309513 Number of even parts in the partitions of n into 3 parts.

%I A309513 #27 Jun 18 2025 07:32:26
%S A309513 0,0,0,0,1,2,5,4,7,8,12,12,18,18,24,24,31,32,41,40,49,50,60,60,72,72,
%T A309513 84,84,97,98,113,112,127,128,144,144,162,162,180,180,199,200,221,220,
%U A309513 241,242,264,264,288,288,312,312,337,338,365,364,391,392,420,420
%N A309513 Number of even parts in the partitions of n into 3 parts.
%H A309513 Ray Chandler, <a href="/A309513/b309513.txt">Table of n, a(n) for n = 0..1000</a>
%H A309513 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A309513 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1,1,-1,0,0,-1,1).
%F A309513 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (((i-1) mod 2) + ((j-1) mod 2) + ((n-i-j-1) mod 2)). [Corrected by _Georg Fischer_, Mar 11 2025]
%F A309513 From _Colin Barker_, Aug 06 2019: (Start)
%F A309513 G.f.: x^4*(1 + x + 3*x^2 - x^3 + 2*x^4) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
%F A309513 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>10. (End)
%F A309513 a(n) = 3*A069905(n) - A309511(n). - _Ray Chandler_, Mar 13 2025
%e A309513 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A309513                                                           1+1+8
%e A309513                                                    1+1+7  1+2+7
%e A309513                                                    1+2+6  1+3+6
%e A309513                                             1+1+6  1+3+5  1+4+5
%e A309513                                      1+1+5  1+2+5  1+4+4  2+2+6
%e A309513                               1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
%e A309513                        1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
%e A309513          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 A309513 -----------------------------------------------------------------------
%e A309513   n  |     3      4      5      6      7      8      9     10      ...
%e A309513 -----------------------------------------------------------------------
%e A309513 a(n) |     0      1      2      5      4      7      8     12      ...
%e A309513 -----------------------------------------------------------------------
%t A309513 Table[Sum[Sum[Mod[i - 1, 2] + Mod[j - 1, 2] + Mod[n - i - j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%Y A309513 Cf. A069905, A128012, A309511.
%K A309513 nonn,easy
%O A309513 0,6
%A A309513 _Wesley Ivan Hurt_, Aug 05 2019