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 1-4 of 4 results.

A370640 Number of maximal subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

1, 1, 1, 3, 3, 8, 17, 32, 32, 77, 144, 242, 383, 580, 843, 1201, 1201, 2694, 4614, 7096, 10219, 14186, 19070, 25207, 32791, 42160, 53329, 66993, 82811, 101963, 124381, 151286, 151286, 324695, 526866, 764438, 1038089, 1358129, 1725921, 2154668, 2640365, 3202985
Offset: 0

Views

Author

Gus Wiseman, Mar 10 2024

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Also choices of A070939(n) elements of {1..n} such that it is possible to choose a different binary index of each.

Examples

			The a(0) = 1 through a(6) = 17 subsets:
  {}  {1}  {1,2}  {1,2}  {1,2,4}  {1,2,4}  {1,2,4}
                  {1,3}  {1,3,4}  {1,2,5}  {1,2,5}
                  {2,3}  {2,3,4}  {1,3,4}  {1,2,6}
                                  {1,3,5}  {1,3,4}
                                  {2,3,4}  {1,3,5}
                                  {2,3,5}  {1,3,6}
                                  {2,4,5}  {1,4,6}
                                  {3,4,5}  {1,5,6}
                                           {2,3,4}
                                           {2,3,5}
                                           {2,3,6}
                                           {2,4,5}
                                           {2,5,6}
                                           {3,4,5}
                                           {3,4,6}
                                           {3,5,6}
                                           {4,5,6}
The a(0) = 1 through a(6) = 17 set-systems:
    {1}  {1}{2}  {1}{2}   {1}{2}{3}   {1}{2}{3}    {1}{2}{3}
                 {1}{12}  {1}{12}{3}  {1}{12}{3}   {1}{12}{3}
                 {2}{12}  {2}{12}{3}  {1}{2}{13}   {1}{2}{13}
                                      {2}{12}{3}   {1}{2}{23}
                                      {2}{3}{13}   {1}{3}{23}
                                      {1}{12}{13}  {2}{12}{3}
                                      {12}{3}{13}  {2}{3}{13}
                                      {2}{12}{13}  {1}{12}{13}
                                                   {1}{12}{23}
                                                   {1}{13}{23}
                                                   {12}{3}{13}
                                                   {12}{3}{23}
                                                   {2}{12}{13}
                                                   {2}{12}{23}
                                                   {2}{13}{23}
                                                   {3}{13}{23}
                                                   {12}{13}{23}

Crossrefs

Dominated by A357812.
The version for set-systems is A368601, max of A367902 (complement A367903).
For prime indices we have A370585, with n A370590, see also A370591.
This is the maximal case of A370636 (complement A370637).
The case of a unique choice is A370638.
The case containing n is A370641, non-maximal A370639.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A133686, A326031, A326702, A367905, A367909, A367912, A368109, A368110, A370592, A370642.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]
  • PARI
    lista(nn) = my(b, m=Map(Mat([[[]], 1])), t, u, v, w, z); for(n=0, nn, t=Mat(m)~; b=Vecrev(binary(n)); u=select(i->b[i], [1..#b]); for(i=1, #t, v=t[1, i]; w=List([]); for(j=1, #v, for(k=1, #u, if(!setsearch(v[j], u[k]), listput(w, setunion(v[j], [u[k]]))))); w=Set(w); if(#w, z=0; mapisdefined(m, w, &z); mapput(m, w, z+t[2, i]))); print1(mapget(m, [[1..#b]]), ", ")); \\ Jinyuan Wang, Mar 28 2025

Extensions

More terms from Jinyuan Wang, Mar 28 2025

A370641 Number of maximal subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 9, 15, 32, 45, 67, 98, 141, 197, 263, 358, 1201, 1493, 1920, 2482, 3123, 3967, 4884, 6137, 7584, 9369, 11169, 13664, 15818, 19152, 22418, 26905, 151286, 173409, 202171, 237572, 273651, 320040, 367792, 428747, 485697, 562620, 637043, 734738, 815492
Offset: 0

Views

Author

Gus Wiseman, Mar 11 2024

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Also choices of A070939(n) elements of {1..n} containing n such that it is possible to choose a different binary index of each.

Examples

			The a(0) = 0 through a(7) = 15 subsets:
  .  {1}  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {1,2,7}
                 {2,3}  {1,3,4}  {1,3,5}  {1,3,6}  {1,3,7}
                        {2,3,4}  {2,3,5}  {1,4,6}  {1,4,7}
                                 {2,4,5}  {1,5,6}  {1,5,7}
                                 {3,4,5}  {2,3,6}  {1,6,7}
                                          {2,5,6}  {2,3,7}
                                          {3,4,6}  {2,4,7}
                                          {3,5,6}  {2,5,7}
                                          {4,5,6}  {2,6,7}
                                                   {3,4,7}
                                                   {3,5,7}
                                                   {3,6,7}
                                                   {4,5,7}
                                                   {4,6,7}
                                                   {5,6,7}

Crossrefs

A version for set-systems is A368601.
For prime indices we have A370590, without n A370585, see also A370591.
This is the maximal case of A370636 requiring n, complement A370637.
This is the maximal case of A370639, complement A370589.
Without requiring n we have A370640.
Dominated by A370819.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
Cf. A133686, A326031, A326702, A357812, A367905, A368100, A368109, A370586, A370638, A370642.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}],MemberQ[#,n] && Length[Union[Sort/@Select[Tuples[bpe/@#], UnsameQ@@#&]]]>0&]],{n,0,25}]

Extensions

More terms from Jinyuan Wang, Mar 28 2025

A357927 Number of subsets of [n] in which exactly half of the elements are Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 15, 35, 56, 126, 252, 462, 792, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, 2629575, 3365856, 4272048, 18156204, 23535820, 30260340, 38608020, 48903492
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2022

Keywords

Examples

			a(6) = 15: {}, {1,4}, {1,6}, {2,4}, {2,6}, {3,4}, {3,6}, {4,5}, {5,6}, {1,2,4,6}, {1,3,4,6}, {1,4,5,6}, {2,3,4,6}, {2,4,5,6}, {3,4,5,6}.

Links

Crossrefs

Cf. A000045, A037031, A072649, A102366, A180272, A357812.

Programs

  • Maple
    f:= proc(n) option remember; `if`(n=0, 0, f(n-1)+
     `if`((t-> ormap(issqr, [t-4, t+4]))(5*n^2), 1, 0))
    end:
a:= n-> binomial(n, f(n)):
seq(a(n), n=0..38);
  • Mathematica
    f[n_] := Module[{j}, For[j = Floor@Log[GoldenRatio, n], Fibonacci[j+1] <= n, j++]; j-1];
a[n_] := If[n == 0, 1, Binomial[n, f[n]]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Nov 17 2022 *)

Formula

a(n) = binomial(n,A072649(n)).
a(n) = Sum_{j>=0} binomial(A072649(n),j)*binomial(n-A072649(n),j).

A370819 Number of subsets of {1..n-1} whose cardinality is one less than the length of the binary expansion of n; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 10, 15, 35, 56, 84, 120, 165, 220, 286, 364, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757, 658008, 749398, 850668
Offset: 0

Views

Author

Gus Wiseman, Mar 11 2024

Keywords

Examples

			The a(1) = 1 through a(7) = 15 subsets:
  {}  {1}  {1}  {1,2}  {1,2}  {1,2}  {1,2}
           {2}  {1,3}  {1,3}  {1,3}  {1,3}
                {2,3}  {1,4}  {1,4}  {1,4}
                       {2,3}  {1,5}  {1,5}
                       {2,4}  {2,3}  {1,6}
                       {3,4}  {2,4}  {2,3}
                              {2,5}  {2,4}
                              {3,4}  {2,5}
                              {3,5}  {2,6}
                              {4,5}  {3,4}
                                     {3,5}
                                     {3,6}
                                     {4,5}
                                     {4,6}
                                     {5,6}

Crossrefs

The version without subtracting one is A357812.
Dominates A370641, see also A370640.
A007318 counts subsets by cardinality.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A029837, A113473, A326031, A367905, A368109, A370636.

Programs

  • Mathematica
    Table[If[n==0,0,Binomial[n-1,IntegerLength[n,2]-1]],{n,0,15}]

Formula

a(n) = binomial(n - 1, A029837(n+1) - 1) = binomial(n - 1, A113473(n) - 1) = binomial(n - 1, A070939(n) - 1) for n > 0.
Showing 1-4 of 4 results.

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