A373579 -id:A373579 - cp's OEIS frontend

A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).

2, 3, 4, 4, 3, 5, 5, 3, 5, 4, 6, 6, 3, 6, 4, 6, 5, 6, 5, 4, 7, 7, 3, 7, 4, 7, 5, 7, 5, 4, 7, 6, 7, 6, 4, 7, 6, 5, 8, 8, 3, 8, 4, 8, 5, 8, 5, 4, 8, 6, 8, 6, 4, 8, 6, 5, 8, 7, 8, 7, 4, 8, 7, 5, 8, 7, 6, 8, 7, 6, 5, 9, 9, 3, 9, 4, 9, 5, 9, 5, 4, 9, 6, 9, 6, 4, 9, 6, 5

Offset: 1

Paolo Xausa, Jun 09 2024

A strong Schreier set is a subset of the positive integers with cardinality less than the minimum element in the set (see Chu link).
Each term k of 2*A371176 can be put into a one-to-one correspondence with a strong Schreier set by interpreting the 1-based position of the ones in the binary expansion of k (where position 1 corresponds to the least significant bit) as the elements of the corresponding strong Schreier set.
Arranging the elements in each set in decreasing order results in the sets being listed in lexicographical order (see example). Cf. A373579 for the elements arranged in increasing order.
The number of sets having maximum element m is A000045(m-1).

			Triangle begins:
                                        Corresponding
   n  A371176(2*n)  bin(A371176(2*n))   strong Schreier set
                                        (this sequence)
  ---------------------------------------------------------
   1        2               10          {2}
   2        4              100          {3}
   3        8             1000          {4}
   4       12             1100          {4, 3}
   5       16            10000          {5}       Sets are
   6       20            10100          {5, 3}    lexicographically
   7       24            11000          {5, 4}    ordered
   8       32           100000          {6}
   9       36           100100          {6, 3}
  10       40           101000          {6, 4}
  11       48           110000          {6, 5}
  12       56           111000          {6, 5, 4}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Hùng Việt Chu, The Fibonacci Sequence and Schreier-Zeckendorf Sets, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.

Subsequence of A373345.

Cf. A000045, A007895 (conjectured row lengths), A371176, A373556, A373579, A373853 (row sums).

Join[{{2}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[4, 400, 4], DigitCount[#, 2, 1] < IntegerExponent[#, 2] + 1 &]]]

T(n,k) = A373345(n,k) + 1.

A373359 Irregular triangle read by rows where row n lists (in increasing order) the elements of the Schreier set encoded by A371176(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 2, 4, 3, 4, 5, 2, 5, 3, 5, 4, 5, 3, 4, 5, 6, 2, 6, 3, 6, 4, 6, 3, 4, 6, 5, 6, 3, 5, 6, 4, 5, 6, 7, 2, 7, 3, 7, 4, 7, 3, 4, 7, 5, 7, 3, 5, 7, 4, 5, 7, 6, 7, 3, 6, 7, 4, 6, 7, 5, 6, 7, 4, 5, 6, 7, 8, 2, 8, 3, 8, 4, 8, 3, 4, 8, 5, 8, 3, 5, 8, 4, 5, 8, 6, 8

Offset: 1

Paolo Xausa, Jun 04 2024

See A373345 (where elements in each set are listed in decreasing order) for more information.

			Triangle begins:
                                   Corresponding Schreier
   n  A371176(n)  bin(A371176(n))  set (this sequence)
  -------------------------------------------------------
   1      1              1         {1}
   2      2             10         {2}
   3      4            100         {3}
   4      6            110         {2, 3}
   5      8           1000         {4}
   6     10           1010         {2, 4}
   7     12           1100         {3, 4}
   8     16          10000         {5}
   9     18          10010         {2, 5}
  10     20          10100         {3, 5}
  11     24          11000         {4, 5}
  12     28          11100         {3, 4, 5}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.

Cf. A371176, A373345, A373558, A373579.

Cf. A007895 (conjectured row lengths), A373346 (row sums), A373347.

Join[{{1}}, Map[PositionIndex[Reverse[IntegerDigits[#, 2]]][1] &, Select[Range[2, 200, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]]]

T(n,k) = A373579(n,k) - 1.

A373558 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in increasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 3, 4, 5, 2, 6, 3, 4, 6, 3, 5, 6, 2, 7, 3, 4, 7, 3, 5, 7, 3, 6, 7, 4, 5, 6, 7, 2, 8, 3, 4, 8, 3, 5, 8, 3, 6, 8, 4, 5, 6, 8, 3, 7, 8, 4, 5, 7, 8, 4, 6, 7, 8, 2, 9, 3, 4, 9, 3, 5, 9, 3, 6, 9, 4, 5, 6, 9, 3, 7, 9, 4, 5, 7, 9, 4, 6, 7, 9, 3, 8, 9

Offset: 1

Paolo Xausa, Jun 10 2024

See A373556 (where elements in each set are listed in decreasing order) for more information.

			Triangle begins:
                                           Corresponding
   n  2*A355489(n-1)  bin(2*A355489(n-1))  maximal Schreier set
                                           (this sequence)
  ---------------------------------------------------------------
   1                                       {1}
   2         6                 110         {2, 3}
   3        10                1010         {2, 4}
   4        18               10010         {2, 5}
   5        28               11100         {3, 4, 4}
   6        34              100010         {2, 6}
   7        44              101100         {3, 4, 6}
   8        52              110100         {3, 5, 6}
   9        66             1000010         {2, 7}
  10        76             1001100         {3, 4, 7}
  11        84             1010100         {3, 5, 7}
  12       100             1100100         {3, 6, 7}
  13       120             1111000         {4, 5, 6, 7}
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10003 (rows 1..1892 of the triangle, flattend).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
Hùng Việt Chu, The Fibonacci Sequence and Schreier-Zeckendorf Sets, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.

Subsequence of A373359.

Cf. A143299 (conjectured row lengths), A355489, A373556, A373579, A373854 (row sums).

Join[{{1}}, Map[PositionIndex[Reverse[IntegerDigits[#, 2]]][1] &, Select[Range[2, 500, 2], DigitCount[#, 2, 1] == IntegerExponent[#, 2] + 1 &]]]

cp's OEIS Frontend

A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A373359 Irregular triangle read by rows where row n lists (in increasing order) the elements of the Schreier set encoded by A371176(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A373558 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in increasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica