A373854 -id:A373854 - cp's OEIS frontend

A373556 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in decreasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

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

Offset: 1

Paolo Xausa, Jun 09 2024

A maximal Schreier set is a subset of the positive integers with cardinality equal to the minimum element in the set (see Chu link).
For n >= 2, each term k = 2*A355489(n-1) can be put into a one-to-one correspondence with a maximal 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 maximal Schreier set.
See A373558 for the elements in each set arranged in increasing order.
The number of sets having maximum element m (for m >= 2) is A000045(m-2).

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

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 A373345.

Cf. A000045, A143299 (conjectured row lengths), A355489, A373557, A373558, A373854 (row sums).

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

A373853 Row sums of A373557.

Original entry on oeis.org

2, 3, 4, 7, 5, 8, 9, 6, 9, 10, 11, 15, 7, 10, 11, 12, 16, 13, 17, 18, 8, 11, 12, 13, 17, 14, 18, 19, 15, 19, 20, 21, 26, 9, 12, 13, 14, 18, 15, 19, 20, 16, 20, 21, 22, 27, 17, 21, 22, 23, 28, 24, 29, 30, 10, 13, 14, 15, 19, 16, 20, 21, 17, 21, 22, 23, 28, 18, 22

Offset: 1

Paolo Xausa, Jun 19 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A007895, A373346, A373557, A373854.

Join[{2}, Map[Total[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[4, 1000, 4], DigitCount[#, 2, 1] < IntegerExponent[#, 2] + 1 &]]]

Conjecture: a(n) = A373346(n) + A007895(n).

cp's OEIS Frontend

A373556 Irregular triangle read by rows: T(1,1) = 1 and, for n >= 2, row n lists (in decreasing order) the elements of the maximal Schreier set encoded by 2*A355489(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

A373853 Row sums of A373557.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula