cp's OEIS Frontend

A307872 Sum of the smallest parts in the partitions of n into 3 parts.

%I A307872 #27 Jan 14 2024 11:25:40
%S A307872 0,0,1,1,2,4,5,7,11,13,17,23,27,33,42,48,57,69,78,90,106,118,134,154,
%T A307872 170,190,215,235,260,290,315,345,381,411,447,489,525,567,616,658,707,
%U A307872 763,812,868,932,988,1052,1124,1188,1260,1341,1413,1494,1584,1665
%N A307872 Sum of the smallest parts in the partitions of n into 3 parts.
%H A307872 Michael De Vlieger, <a href="/A307872/b307872.txt">Table of n, a(n) for n = 1..3000</a>
%H A307872 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A307872 a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} k.
%F A307872 Conjectures from _Colin Barker_, May 02 2019: (Start)
%F A307872 G.f.: x^3 / ((1 - x)^4*(1 + x)*(1 + x + x^2)^2).
%F A307872 a(n) = a(n-1) + a(n-2) + a(n-3) - 2*a(n-4) - 2*a(n-5) + a(n-6) + a(n-7) + a(n-8) - a(n-9) for n > 9.
%F A307872 (End)
%F A307872 a(n) = ((-1)^n*(-1+(-1)^r)+2*r*((-1)^(n+r)+(1+r)*(1+2*n-4*r)))/16, where r = floor(n/3). - _Wesley Ivan Hurt_, Oct 24 2021
%e A307872 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A307872                                                           1+1+8
%e A307872                                                    1+1+7  1+2+7
%e A307872                                                    1+2+6  1+3+6
%e A307872                                             1+1+6  1+3+5  1+4+5
%e A307872                                      1+1+5  1+2+5  1+4+4  2+2+6
%e A307872                               1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
%e A307872                        1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
%e A307872          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 A307872 -----------------------------------------------------------------------
%e A307872   n  |     3      4      5      6      7      8      9     10      ...
%e A307872 -----------------------------------------------------------------------
%e A307872 a(n) |     1      1      2      4      5      7     11     13      ...
%e A307872 -----------------------------------------------------------------------
%t A307872 Table[Sum[Sum[k, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%t A307872 Table[Total[IntegerPartitions[n,{3}][[;;,-1]]],{n,100}] (* _Harvey P. Dale_, Jan 14 2024 *)
%o A307872 (PARI) a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, k)); \\ _Michel Marcus_, May 02 2019
%Y A307872 Cf. A069905.
%K A307872 nonn,easy
%O A307872 1,5
%A A307872 _Wesley Ivan Hurt_, May 02 2019