cp's OEIS Frontend

A309511 Number of odd parts in the partitions of n into 3 parts.

%I A309511 #16 Mar 13 2025 11:55:21
%S A309511 0,0,0,3,2,4,4,8,8,13,12,18,18,24,24,33,32,40,40,50,50,61,60,72,72,84,
%T A309511 84,99,98,112,112,128,128,145,144,162,162,180,180,201,200,220,220,242,
%U A309511 242,265,264,288,288,312,312,339,338,364,364,392,392,421,420
%N A309511 Number of odd parts in the partitions of n into 3 parts.
%H A309511 Ray Chandler, <a href="/A309511/b309511.txt">Table of n, a(n) for n = 0..1000</a>
%H A309511 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A309511 <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 A309511 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} ((i mod 2) + (j mod 2) + ((n-i-j) mod 2)).
%F A309511 From _Colin Barker_, Aug 06 2019: (Start)
%F A309511 G.f.: x^3*(3 - x + 2*x^2 + x^4 + x^5) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
%F A309511 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.
%F A309511 (End)
%F A309511 a(n) = 3*A069905(n) - A309513(n). - _Ray Chandler_, Mar 13 2025
%e A309511 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A309511                                                           1+1+8
%e A309511                                                    1+1+7  1+2+7
%e A309511                                                    1+2+6  1+3+6
%e A309511                                             1+1+6  1+3+5  1+4+5
%e A309511                                      1+1+5  1+2+5  1+4+4  2+2+6
%e A309511                               1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
%e A309511                        1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
%e A309511          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 A309511 -----------------------------------------------------------------------
%e A309511   n  |     3      4      5      6      7      8      9     10      ...
%e A309511 -----------------------------------------------------------------------
%e A309511 a(n) |     3      2      4      4      8      8     13     12      ...
%e A309511 -----------------------------------------------------------------------
%t A309511 Table[Sum[Sum[Mod[i, 2] + Mod[j, 2] + Mod[n - i - j, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t A309511 Table[Count[Flatten[IntegerPartitions[n,{3}]],_?OddQ],{n,0,60}] (* _Harvey P. Dale_, Jan 16 2022 *)
%K A309511 nonn
%O A309511 0,4
%A A309511 _Wesley Ivan Hurt_, Aug 05 2019