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%I #24 Jul 04 2023 15:53:50
%S 0,1,3,3,8,11,17,23,36,48,69,88,125,157,212,271,356,445,574,711,906,
%T 1118,1400,1711,2125,2583,3171,3828,4666,5604,6777,8095,9730,11567,
%U 13815,16357,19429,22910,27077,31801,37432,43802,51338,59871,69914,81273,94562
%N Number of parts in all partitions of n in which no part occurs more than twice.
%H Vaclav Kotesovec, <a href="/A185350/b185350.txt">Table of n, a(n) for n = 0..11410</a> (terms 0..1000 from Alois P. Heinz)
%F a(n) = Sum_{k>=0} k*A209318(n,k).
%F a(n) ~ log(3) * exp(2*Pi*sqrt(n)/3) / (2*Pi*n^(1/4)). - _Vaclav Kotesovec_, May 26 2018
%e a(6) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
%e a(7) = 23: [7], [6,1], [5,2], [4,3], [5,1,1], [4,2,1], [3,3,1], [3,2,2], [3,2,1,1].
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 2))))
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=0..50);
%t b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[b[n - i j, i - 1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]] j}, {j, 0, Min[n/i, k]} ] ] ];
%t a[n_] := b[n, n, 2][[2]];
%t a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%t Table[Length[Flatten[Select[IntegerPartitions[n],Max[Length/@Split[#]]<3&]]],{n,0,50}] (* _Harvey P. Dale_, Jul 04 2023 *)
%Y Column k=2 of A210485.
%Y Cf. A015723, A117148, A209318.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jan 21 2013
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