This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A238213 #12 Jul 05 2025 08:45:43
%S A238213 0,0,0,0,0,0,1,0,0,1,1,2,2,2,2,3,4,5,6,7,8,10,12,15,17,20,23,27,33,38,
%T A238213 44,51,59,68,79,91,104,119,136,155,178,202,230,261,296,335,379,428,
%U A238213 483,544,612,688,773,867,972,1088,1217,1360,1518,1693,1887
%N A238213 The total number of 6's in all partitions of n into an odd number of distinct parts.
%C A238213 The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).
%H A238213 Andrew Howroyd, <a href="/A238213/b238213.txt">Table of n, a(n) for n = 0..1000</a>
%F A238213 a(n) = Sum_{j=1..round(n/12)} A067661(n-(2*j-1)*6) - Sum_{j=1..floor(n/12)} A067659(n-12*j).
%F A238213 G.f.: (1/2)*(x^6/(1+x^6))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^6/(1-x^6))*(Product_{n>=1} 1 - x^n).
%F A238213 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238213 a(12) = 2 because the partitions in question are: 6+5+1, 6+4+2.
%t A238213 nmax = 100; With[{k=6}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 05 2025 *)
%Y A238213 Column k=6 of A238450.
%Y A238213 Cf. A067659, A067661.
%K A238213 nonn
%O A238213 0,12
%A A238213 _Mircea Merca_, Feb 20 2014
%E A238213 Terms a(51) and beyond from _Andrew Howroyd_, Apr 29 2020