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

0,6

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

Links

Table of n, a(n) for n=0..86.

Example

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)

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}]

Crossrefs

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

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

Sequence in context: A289499 A215904 A083552 * A332207 A339929 A176835

Adjacent sequences: A326031 A326032 A326033 * A326035 A326036 A326037

Keyword

nonn

Author

Gus Wiseman, Jun 04 2019

Status

approved