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.

A364671 Number of subsets of {1..n} containing all of their own first differences.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 23, 34, 58, 96, 171, 302, 565, 1041, 1969, 3719, 7105, 13544, 25999, 49852, 95949, 184658, 356129, 687068, 1327540, 2566295, 4966449, 9617306, 18640098, 36150918, 70166056, 136272548, 264844111, 515036040, 1002211421, 1951345157, 3801569113
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2023

Keywords

Examples

			The subset {1,2,4,5,10,14} has differences (1,2,1,5,4) so is counted under a(14).
The a(0) = 1 through a(5) = 14 subsets:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {2}    {2}      {2}        {2}
           {1,2}  {3}      {3}        {3}
                  {1,2}    {4}        {4}
                  {1,2,3}  {1,2}      {5}
                           {2,4}      {1,2}
                           {1,2,3}    {2,4}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}

Links

Crossrefs

For differences of all strict pairs we have A054519, for partitions A007862.
For "disjoint" instead of "subset" we have A364463, partitions A363260.
For "non-disjoint" we have A364466, partitions A364467 (strict A364536).
The complement is counted by A364672, partitions A364673, A364674, A364675.
First differences of terms are A364752, complement A364753.
Cf. A151897, A196723, A237667, A237668, A325325, A326083, A363225, A364345, A364464, A364537.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Differences[#]]&]], {n,0,10}]

Extensions

More terms from Rémy Sigrist, Aug 06 2023

A364672 Number of subsets of {1..n} not containing all of their own first differences.

Original entry on oeis.org

0, 0, 0, 2, 6, 18, 41, 94, 198, 416, 853, 1746, 3531, 7151, 14415, 29049, 58431, 117528, 236145, 474436, 952627, 1912494, 3838175, 7701540, 15449676, 30988137, 62142415, 124600422, 249795358, 500719994, 1003575768, 2011211100, 4030123185, 8074898552, 16177657763, 32408393211, 64917907623
Offset: 0

Views

Author

Gus Wiseman, Aug 05 2023

Keywords

Examples

			The a(0) = 0 through a(5) = 18 subsets:
  .  .  .  {1,3}  {1,3}    {1,3}
           {2,3}  {1,4}    {1,4}
                  {2,3}    {1,5}
                  {3,4}    {2,3}
                  {1,3,4}  {2,5}
                  {2,3,4}  {3,4}
                           {3,5}
                           {4,5}
                           {1,2,5}
                           {1,3,4}
                           {1,3,5}
                           {1,4,5}
                           {2,3,4}
                           {2,3,5}
                           {2,4,5}
                           {3,4,5}
                           {1,3,4,5}
                           {2,3,4,5}

Crossrefs

For disjunction instead of containment we have A364463, partitions A363260.
For overlap we have A364466, partitions A364467 (strict A364536).
The complement is counted by A364671, partitions A364673, A364674, A364675.
First differences of terms are A364753, complement A364752.
Cf. A007862, A054519, A151897, A237667, A325325, A326083, A363225, A364345, A364464, A364537.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],!SubsetQ[#,Differences[#]]&]],{n,0,10}]

Formula

a(n) = 2^n - A364671(n). - Andrew Howroyd, Jan 27 2024

Extensions

a(21) onwards (using A364671) added by Andrew Howroyd, Jan 27 2024

A364753 Number of subsets of {1..n} containing n but not containing all first differences.

Original entry on oeis.org

0, 0, 0, 2, 4, 12, 23, 53, 104, 218, 437, 893, 1785, 3620, 7264, 14634, 29382, 59097, 118617, 238291, 478191, 959867, 1925681, 3863365, 7748136, 15538461, 31154278, 62458007, 125194936, 250924636, 502855774, 1007635332, 2018912085, 4044775367, 8102759211, 16230735448, 32509514412, 65110826347
Offset: 0

Views

Author

Gus Wiseman, Aug 06 2023

Keywords

Comments

In other words, subsets containing both n and some element that is not the difference of two consecutive elements.

Examples

			The a(1) = 0 through a(6) = 23 subsets:
  .  .  {1,3}  {1,4}    {1,5}      {1,6}
        {2,3}  {3,4}    {2,5}      {2,6}
               {1,3,4}  {3,5}      {4,6}
               {2,3,4}  {4,5}      {5,6}
                        {1,2,5}    {1,2,6}
                        {1,3,5}    {1,3,6}
                        {1,4,5}    {1,4,6}
                        {2,3,5}    {1,5,6}
                        {2,4,5}    {2,3,6}
                        {3,4,5}    {2,5,6}
                        {1,3,4,5}  {3,4,6}
                        {2,3,4,5}  {3,5,6}
                                   {4,5,6}
                                   {1,2,5,6}
                                   {1,3,4,6}
                                   {1,3,5,6}
                                   {1,4,5,6}
                                   {2,3,4,6}
                                   {2,3,5,6}
                                   {2,4,5,6}
                                   {3,4,5,6}
                                   {1,3,4,5,6}
                                   {2,3,4,5,6}

Crossrefs

Partial sums are A364672, complement A364671.
The complement is counted by A364752.
A054519 counts subsets containing differences, A326083 containing sums.
A364463 counts subsets disjoint from differences, complement A364466.
A364673, A364674, A364675 count partitions containing differences.
Cf. A151897, A196723, A237668, A325325, A364345, A364464, A364537.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!SubsetQ[#,Differences[#]]&]],{n,0,10}]

Extensions

More terms from Giorgos Kalogeropoulos, Aug 07 2023
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.