A373359 -id:A373359 - cp's OEIS frontend

A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

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

Offset: 1

Paolo Xausa, Jun 01 2024

A Schreier set is a subset of the positive integers with cardinality less than or equal to the minimum element in the set.
Each term k of A371176 can be put into a one-to-one correspondence with a 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 Schreier set (see A371176 and Bird link).
See A373359 for the elements in each set arranged in increasing order.
The number of sets having maximum element m is A000045(m).

			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         {3, 2}
   5      8           1000         {4}
   6     10           1010         {4, 2}
   7     12           1100         {4, 3}
   8     16          10000         {5}
   9     18          10010         {5, 2}
  10     20          10100         {5, 3}
  11     24          11000         {5, 4}
  12     28          11100         {5, 4, 3}
  ...

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. A000045, A371176, A373359, A373556, A373557.

Cf. A007895 (conjectured row lengths), A072649 (first column), A373346 (row sums), A373347.

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

T(n,k) = A373557(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 &]]]

A373579 Irregular triangle read by rows where row n lists (in increasing order) the elements of the strong Schreier set encoded by A371176(2*n).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jun 10 2024

See A373557 (where elements in each set are listed in decreasing order) for more information.

			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          {3, 4}
   5       16            10000          {5}
   6       20            10100          {3, 5}
   7       24            11000          {4, 5}
   8       32           100000          {6}
   9       36           100100          {3, 6}
  10       40           101000          {4, 6}
  11       48           110000          {5, 6}
  12       56           111000          {4, 5, 6}
  ...

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 A373359.

Cf. A007895 (conjectured row lengths), A371176, A373557, A373558, A373853 (row sums).

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

T(n,k) = A373359(n,k) + 1.

cp's OEIS Frontend

A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

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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).

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A373579 Irregular triangle read by rows where row n lists (in increasing order) the elements of the strong Schreier set encoded by A371176(2*n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula