A053538 -id:A053538 - cp's OEIS frontend

A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 3, 4, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 3, 3, 4, 5, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2

Offset: 0

Gus Wiseman, Jul 02 2025

nonn

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.

			The binary indices of 27 are {1,2,4,5}, with maximal sparse subsets {{1,4},{1,5},{2,4},{2,5}}, so a(27) = 4.

For subsets of {1..n} we get A000931 (shifted), maximal case of A000045 (shifted).
This is the maximal case of A245564.
The greatest number whose binary indices are one of these subsets is A374356.
For prime instead of binary indices we have A385215, maximal case of A166469.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A202064 counts subsets containing n with k maximal runs.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A044813, A053538, A119900, A202023, A208342, A384177, A384890.

spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
maximize[sys_]:=Complement@@Prepend[Most[Subsets[#]]&/@sys,sys];
Table[Length[maximize[spars[bpe[n]]]],{n,0,100}]

A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 207, 394, 739, 1398, 2687, 5152, 9891, 19128, 37039, 71754, 139459, 271522, 528999, 1032308, 2017291, 3945186, 7723203, 15134440, 29679407, 58245068, 114389683, 224796210, 442021743, 869658304, 1711914351, 3371515306

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

Also the number of subsets of {1..n} with the same number of adjacent elements increasing by 1 as adjacent elements increasing by more than 1.

			The set {2,3,5,6,8} has maximal runs ((2,3),(5,6),(8)) and maximal anti-runs ((2),(3,5),(6,8)) so is counted under a(8).
The a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}   {}       {}       {}
      {1}  {1}  {1}  {1}      {1}      {1}
           {2}  {2}  {2}      {2}      {2}
                {3}  {3}      {3}      {3}
                     {4}      {4}      {4}
                     {1,2,4}  {5}      {5}
                     {1,3,4}  {1,2,4}  {6}
                              {1,2,5}  {1,2,4}
                              {1,3,4}  {1,2,5}
                              {1,4,5}  {1,2,6}
                              {2,3,5}  {1,3,4}
                              {2,4,5}  {1,4,5}
                                       {1,5,6}
                                       {2,3,5}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}

Christian Sievers, Table of n, a(n) for n = 0..3328

The LHS is counted by A034839 (for partitions A384881, strict A116674), rank statistic A069010.
The case containing n + 1 is A217615.
The RHS is counted by A384893 or A210034 (for partitions A268193, strict A384905), rank statistic A384890.
Subsets of this type are ranked by A385575.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000071, A000079, A001629, A010027, A053538, A384177, A384879, A384889.

a:= proc(n) option remember; `if`(n<5, [1, 2, 3, 4, 7][n+1], ((3*n-4)*a(n-1)-
      (3*n-5)*a(n-2)+(5*n-12)*a(n-3)-2*(4*n-11)*a(n-4)+4*(n-3)*a(n-5))/(n-1))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 06 2025

Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2==#1+1&]]==Length[Split[#,#2!=#1+1&]]&]],{n,0,10}]

a(n)=polcoef([1,1,1]*[x,0,0;x,x^2,1;0,x,x]^n*[1,0,0]~,n) \\ Christian Sievers, Jul 06 2025

Let M be the matrix [1,0,0; 1,x,1/x; 0,1,1]. Then a(n) is the sum of the constant terms of the entries in the left column of M^n. - Christian Sievers, Jul 06 2025

a(21) and beyond from Christian Sievers, Jul 06 2025

A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 19, 22, 25, 26, 32, 35, 38, 44, 49, 50, 52, 64, 67, 70, 76, 87, 88, 91, 93, 97, 98, 100, 104, 107, 109, 117, 128, 131, 134, 140, 151, 152, 155, 157, 167, 174, 176, 179, 182, 185, 186, 193, 194, 196, 200, 203, 205, 208, 211, 214, 217

Offset: 1

Gus Wiseman, Jul 04 2025

nonn

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.

			The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    8:    1000 ~ {4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   19:   10011 ~ {1,2,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   32:  100000 ~ {6}
   35:  100011 ~ {1,2,6}
   38:  100110 ~ {2,3,6}
   44:  101100 ~ {3,4,6}
   49:  110001 ~ {1,5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   64: 1000000 ~ {7}
   67: 1000011 ~ {1,2,7}
   70: 1000110 ~ {2,3,7}
   76: 1001100 ~ {3,4,7}
   87: 1010111 ~ {1,2,3,5,7}
   88: 1011000 ~ {4,5,7}
   91: 1011011 ~ {1,2,4,5,7}
   93: 1011101 ~ {1,3,4,5,7}
   97: 1100001 ~ {1,6,7}
   98: 1100010 ~ {2,6,7}
  100: 1100100 ~ {3,6,7}

The LHS rank statistic is A069010, counted by A034839 (for partitions A384881, A116674).
The RHS rank statistic is A384890, counted by A384893 (for partitions A268193, A384905).
Subsets of this type are counted by A385572, with n A217615.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000079, A010027, A053538, A210034, A384177, A384879, A384889.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Length[Split[bpe[#],#2==#1+1&]]==Length[Split[bpe[#],#2!=#1+1&]]&]

is_ok(n)=hammingweight(n)==2*hammingweight(bitand(n,n>>1))+1 \\ Christian Sievers, Jul 18 2025

A124448 Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).

Original entry on oeis.org

1, -2, 1, 2, -3, 1, 0, 4, -4, 1, -2, -1, 7, -5, 1, 0, -4, -4, 11, -6, 1, 4, 2, -6, -10, 16, -7, 1, 0, 8, 8, -6, -20, 22, -8, 1, -10, -5, 11, 19, -1, -35, 29, -9, 1, 0, -20, -20, 7, 34, 13, -56, 37, -10, 1, 28, 14, -26, -46, -12, 49, 41, -84

Offset: 0

Paul Barry, Nov 01 2006

Inverse of triangle A106195.
Row sums are A105523 (expansion of 1-xc(-x^2) where c(x) is the g.f. of A000108).
Product of A007318 and A124448 is inverse of A053538.
A124448*A007318 = A106180, as infinite lower triangular matrices. - Philippe Deléham, Oct 16 2007
Triangle T(n,k), read by rows, given by (-2,1,-1,1,-1,1,-1,1,-1,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 09 2011

			Triangle begins
   1;
  -2,   1;
   2,  -3,   1;
   0,   4,  -4,   1;
  -2,  -1,   7,  -5,   1;
   0,  -4,  -4,  11,  -6,   1;
   4,   2,  -6, -10,  16,  -7,   1;
   0,   8,   8,  -6, -20,  22,  -8,   1;

Cf. A106195, A000045, A164948, A164942.

N=12;
T(n, k)=sum(i=0, n-k, binomial(k, i)*binomial(n-k, i)*2^(n-k-i));
M=matrix(N, N);
for(n=1, N, for(k=1, n, M[n, k]=T(n-1, k-1))); /* A106195 */
A=M^-1;  /* A124448 */
/* for (n=1, N, for(k=1, n, print1(M[n, k], ", "))); */ /* A106195 */
for (n=1, N, for(k=1, n, print1(A[n, k], ", "))); /* A124448 */
/* Joerg Arndt, May 14 2011 */

Edited by N. J. A. Sloane, Dec 29 2011

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A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

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A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

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A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

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A124448 Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).

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