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%I #10 May 14 2021 17:32:29
%S 0,0,0,1,1,2,1,2,2,2,2,3,1,3,2,2,2,3,1,3,2,2,2,3,1,3,2,2,2,3,1,3,2,2,
%T 2,3,1,3,2,2,2,3,1,3,2,2,2,3,1,3,2,2,2,3,1,3,2,2,2,3,1,3,2,2,2,3,1,3,
%U 2,2,2,3,1,3,2,2,2,3,1,3,2,2,2,3,1,3,2
%N Number of knapsack partitions of n with largest part 3.
%C An integer partition is knapsack if every distinct submultiset has a different sum.
%C Appears to repeat the terms (2,2,2,3,1,3) ad infinitum.
%C I computed terms a(n) for n = 0..5000 and (2,2,2,3,1,3) is repeated continuously starting at a(8). - _Fausto A. C. Cariboni_, May 14 2021
%e The initial values count the following partitions:
%e 3: (3)
%e 4: (3,1)
%e 5: (3,2)
%e 5: (3,1,1)
%e 6: (3,3)
%e 7: (3,3,1)
%e 7: (3,2,2)
%e 8: (3,3,2)
%e 8: (3,3,1,1)
%e 9: (3,3,3)
%e 9: (3,2,2,2)
%e 10: (3,3,3,1)
%e 10: (3,3,2,2)
%e 11: (3,3,3,2)
%e 11: (3,3,3,1,1)
%e 11: (3,2,2,2,2)
%e 12: (3,3,3,3)
%e 13: (3,3,3,3,1)
%e 13: (3,3,3,2,2)
%e 13: (3,2,2,2,2,2)
%e 14: (3,3,3,3,2)
%e 14: (3,3,3,3,1,1)
%e 15: (3,3,3,3,3)
%e 15: (3,2,2,2,2,2,2)
%t sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
%t kst[n_]:=Select[IntegerPartitions[n,All,{1,2,3}],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
%t Table[Length[Select[kst[n],Max@@#==3&]],{n,0,30}]
%Y Cf. A002033, A108917, A275972, A276024, A299702.
%Y Cf. A325592, A326015, A326016, A326017, A326018.
%K nonn
%O 0,6
%A _Gus Wiseman_, Jun 04 2019
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