A326034 Number of knapsack partitions of n with largest part 3.

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, 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, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2

Offset: 0

Author

Gus Wiseman, Jun 04 2019

Keywords

• nonn

Comments

An integer partition is knapsack if every distinct submultiset has a different sum.

Appears to repeat the terms (2,2,2,3,1,3) ad infinitum.

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

Examples

			The initial values count the following partitions:
   3: (3)
   4: (3,1)
   5: (3,2)
   5: (3,1,1)
   6: (3,3)
   7: (3,3,1)
   7: (3,2,2)
   8: (3,3,2)
   8: (3,3,1,1)
   9: (3,3,3)
   9: (3,2,2,2)
  10: (3,3,3,1)
  10: (3,3,2,2)
  11: (3,3,3,2)
  11: (3,3,3,1,1)
  11: (3,2,2,2,2)
  12: (3,3,3,3)
  13: (3,3,3,3,1)
  13: (3,3,3,2,2)
  13: (3,2,2,2,2,2)
  14: (3,3,3,3,2)
  14: (3,3,3,3,1,1)
  15: (3,3,3,3,3)
  15: (3,2,2,2,2,2,2)
		

Crossrefs

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325592, A326015, A326016, A326017, A326018.

Programs

Mathematica

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
kst[n_]:=Select[IntegerPartitions[n,All,{1,2,3}],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[kst[n],Max@@#==3&]],{n,0,30}]