A373853 -id:A373853 - 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.

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.

A373854 Row sums of A373556.

Original entry on oeis.org

1, 5, 6, 7, 12, 8, 13, 14, 9, 14, 15, 16, 22, 10, 15, 16, 17, 23, 18, 24, 25, 11, 16, 17, 18, 24, 19, 25, 26, 20, 26, 27, 28, 35, 12, 17, 18, 19, 25, 20, 26, 27, 21, 27, 28, 29, 36, 22, 28, 29, 30, 37, 31, 38, 39, 13, 18, 19, 20, 26, 21, 27, 28, 22, 28, 29, 30

Offset: 1

Paolo Xausa, Jun 19 2024

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

Cf. A373346, A373556, A373853.

Join[{1}, Map[Total[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 2000, 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).

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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).

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A373854 Row sums of A373556.

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