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%I #13 Jun 02 2018 06:38:48
%S 1,0,1,0,2,1,3,2,5,4,8,7,13,12,20,20,31,32,47,50,71,77,105,116,155,
%T 173,225,254,325,369,465,530,660,755,929,1066,1300,1493,1805,2076,
%U 2493,2867,3421,3934,4669,5368,6337,7282,8560,9828,11505,13198,15394,17641,20507,23475
%N Number of partitions of n with no part equal to 1 or 3.
%F a(n) = p(n) - p(n-1) - p(n-3) + p(n-4), where p(n) = A000041(n).
%F G.f.: (1-x-x^3+x^4) / Product_{m>=1} (1-x^m).
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (8*sqrt(3)*n^2). - _Vaclav Kotesovec_, Jun 02 2018
%e a(6)=3 because we have [2,2,2], [2,4], and [6].
%p with(combinat): p := proc (n) options operator, arrow: numbpart(n) end proc: 1, 0, 1, 0, 2, seq(p(n)-p(n-1)-p(n-3)+p(n-4), n = 5 .. 55);
%p G := (1-x)*(1-x^3)/(product(1-x^j, j = 1 .. 65)): Gser := series(G, x = 0, 60): seq(coeff(Gser, x, n), n = 0 .. 55);
%t Table[Count[IntegerPartitions[n],_?(FreeQ[#,1]&&FreeQ[#,3]&)],{n,0,60}] (* _Harvey P. Dale_, Feb 25 2015 *)
%Y Cf. A000041.
%K nonn
%O 0,5
%A _Shanzhen Gao_, Oct 28 2010
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