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%I #8 May 01 2020 17:51:58
%S 0,0,0,0,1,1,0,1,1,1,2,2,2,3,4,5,6,7,9,10,12,15,17,20,24,27,32,38,43,
%T 50,59,67,77,90,102,117,135,153,175,200,226,257,292,330,373,422,475,
%U 535,603,677,760,853,955,1069,1196,1336,1491,1663,1853,2063,2295
%N The total number of 3's in all partitions of n into an even number of distinct parts.
%C 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 Andrew Howroyd, <a href="/A238218/b238218.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{j=1..round(n/6)} A067659(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067661(n-6*j).
%F G.f.: (1/2)*(x^3/(1+x^3))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^3/(1-x^3))*(Product_{n>=1} 1 - x^n).
%e a(13) = 3 because the partitions in question are: 10+3, 7+3+2+1, 5+4+3+1.
%o (PARI) seq(n)={my(A=O(x^(n-2))); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x^3)) - eta(x + A)/(1-x^3))/2, -(n+1))} \\ _Andrew Howroyd_, May 01 2020
%Y Column k=3 of A238451.
%Y Cf. A067659, A067661.
%K nonn
%O 0,11
%A _Mircea Merca_, Feb 20 2014
%E Terms a(51) and beyond from _Andrew Howroyd_, May 01 2020
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