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

1,1

Comments

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).

Links

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.

Formula

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

Example

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}
  ...

Mathematica

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

Crossrefs

Subsequence of A373345.

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

Sequence in context: A160371 A059686 A101083 * A097935 A188940 A286549

Adjacent sequences: A373554 A373555 A373556 * A373558 A373559 A373560

Keyword

nonn,tabf,base,easy

Author

Paolo Xausa, Jun 09 2024

Status

approved