A366320 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} without a subset summing to k.

1, 2, 2, 3, 4, 4, 3, 6, 6, 7, 8, 8, 6, 6, 9, 11, 11, 14, 14, 15, 16, 16, 12, 12, 9, 17, 17, 20, 20, 24, 27, 27, 30, 30, 31, 32, 32, 24, 24, 18, 17, 26, 31, 29, 35, 36, 43, 47, 50, 51, 56, 59, 59, 62, 62, 63

Offset

1,2

Links

Table of n, a(n) for n=1..56.

Example

Triangle begins:
   1
   2  2  3
   4  4  3  6  6  7
   8  8  6  6  9 11 11 14 14 15
  16 16 12 12  9 17 17 20 20 24 27 27 30 30 31
  32 32 24 24 18 17 26 31 29 35 36 43 47 50 51 56 59 59 62 62 63
Row n = 3 counts the following subsets:
  {}     {}     {}   {}     {}     {}
  {2}    {1}    {1}  {1}    {1}    {1}
  {3}    {3}    {2}  {2}    {2}    {2}
  {2,3}  {1,3}       {3}    {3}    {3}
                     {1,2}  {1,2}  {1,2}
                     {2,3}  {1,3}  {1,3}
                                   {2,3}

Mathematica

Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#], k]&]], {n, 8}, {k, n*(n+1)/2}]

Crossrefs

Row lengths are A000217.

The diagonal T(n,n) is A365377, complement A365376.

The complement is counted by A365381.

A000009 counts subsets summing to n.

A000124 counts distinct possible sums of subsets of {1..n}.

A124506 counts combination-free subsets, differences of A326083.

A365046 counts combination-full subsets, differences of A364914.

Cf. A007865, A085489, A093971, A103580, A151897, A326080, A364534, A365073, A365380.

Sequence in context: A086416 A168148 A147968 * A065167 A332298 A300404

Adjacent sequences: A366317 A366318 A366319 * A366321 A366322 A366323

Keyword

nonn,tabf

Author

Gus Wiseman, Oct 12 2023

Status

approved