cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A326020 Number of complete subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 15, 27, 50, 95, 185, 365, 724, 1441, 2873, 5735, 11458, 22902, 45789, 91561, 183102, 366180, 732331, 1464626, 2929209, 5858367, 11716674, 23433277, 46866473, 93732852, 187465596, 374931067, 749861989, 1499723808, 2999447418
Offset: 0

Views

Author

Gus Wiseman, Jun 04 2019

Keywords

Comments

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

Examples

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,3,4}
                                      {1,2,3,5}    {1,2,3,5}
                                      {1,2,4,5}    {1,2,3,6}
                                      {1,2,3,4,5}  {1,2,4,5}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,4,5,6}
                                                   {1,2,3,4,5,6}

Links

Crossrefs

Cf. A002033, A103295, A108917, A126796, A188431, A276024.
Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326021, A326022, A326036.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,0,10}]

Extensions

a(17)-a(34) from Charlie Neder, Jun 05 2019

A326115 Number of maximal double-free subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 12, 12, 12, 12, 24, 24, 32, 32, 64, 64, 64, 64, 128, 128, 192, 192, 384, 384, 384, 384, 768, 768, 960, 960, 1920, 1920, 1920, 1920, 3840, 3840, 5760, 5760, 11520, 11520, 11520, 11520, 23040, 23040, 30720, 30720
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2019

Keywords

Comments

A set is double-free if no element is twice any other element.

Examples

			The a(1) = 1 through a(9) = 6 sets:
  {1}  {1}  {13}  {23}   {235}   {235}   {2357}   {13457}  {134579}
       {2}  {23}  {134}  {1345}  {256}   {2567}   {13578}  {135789}
                                 {1345}  {13457}  {14567}  {145679}
                                 {1456}  {14567}  {15678}  {156789}
                                                  {23578}  {235789}
                                                  {25678}  {256789}

Links

Crossrefs

Cf. A000931, A007865, A050291, A103580, A120641, A308546, A320340, A323092.
Cf. A325865, A326020, A326022, A326083.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,2*#]=={}&]]],{n,0,10}]

Formula

From Charlie Neder, Jun 11 2019: (Start)
a(n) = Product {k < n/2} A000931(8+floor(log_2(n/(2k+1)))).
a(2k+1) = a(2k), a(8k+4) = a(8k+3). (End)

Extensions

a(16)-a(49) from Charlie Neder, Jun 11 2019

A326021 Number of complete subsets of {1..n} with maximum n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 23, 45, 90, 180, 359, 717, 1432, 2862, 5723, 11444, 22887, 45772, 91541, 183078, 366151, 732295, 1464583, 2929158, 5858307, 11716603, 23433196, 46866379, 93732744, 187465471, 374930922, 749861819, 1499723610
Offset: 1

Views

Author

Gus Wiseman, Jun 04 2019

Keywords

Comments

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

Examples

			The a(1) = 1 through a(7) = 12 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,4}    {1,2,3,5}    {1,2,3,6}      {1,2,3,7}
                       {1,2,3,4}  {1,2,4,5}    {1,2,4,6}      {1,2,4,7}
                                  {1,2,3,4,5}  {1,2,3,4,6}    {1,2,3,4,7}
                                               {1,2,3,5,6}    {1,2,3,5,7}
                                               {1,2,4,5,6}    {1,2,3,6,7}
                                               {1,2,3,4,5,6}  {1,2,4,5,7}
                                                              {1,2,4,6,7}
                                                              {1,2,3,4,5,7}
                                                              {1,2,3,4,6,7}
                                                              {1,2,3,5,6,7}
                                                              {1,2,4,5,6,7}
                                                              {1,2,3,4,5,6,7}

Links

Crossrefs

Cf. A002033, A103295, A108917, A126796, A188431, A276024.
Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326022, A326036.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,10}]

Extensions

a(18)-a(34) from Charlie Neder, Jun 05 2019
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.