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

Author

Paolo Xausa, Jun 09 2024

Keywords

• nonn
• tabf
• base
• easy

Comments

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).

Examples

			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}
  ...
		

Links

• 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.

Crossrefs

Subsequence of A373345.

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

Programs

Mathematica

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