%I #30 Sep 13 2024 15:56:52

%S 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,

%T 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,

%U 6,3,9,6,5,4,9,7,3,9,7,5,4,9,7,6,4,9,8,3

%N 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).

%C A maximal Schreier set is a subset of the positive integers with cardinality equal to the minimum element in the set (see Chu link).

%C 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.

%C See A373558 for the elements in each set arranged in increasing order.

%C The number of sets having maximum element m (for m >= 2) is A000045(m-2).

%H Paolo Xausa, <a href="/A373556/b373556.txt">Table of n, a(n) for n = 1..10003</a> (rows 1..1892 of the triangle, flattend).

%H Alistair Bird, <a href="https://outofthenormmaths.wordpress.com/2012/05/13/jozef-schreier-schreier-sets-and-the-fibonacci-sequence/">Jozef Schreier, Schreier sets and the Fibonacci sequence</a>, Out Of The Norm blog, May 13 2012.

%H Hùng Việt Chu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Chu2/chu9.pdf">The Fibonacci Sequence and Schreier-Zeckendorf Sets</a>, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.

%e Triangle begins:

%e Corresponding

%e n 2*A355489(n-1) bin(2*A355489(n-1)) maximal Schreier set

%e (this sequence)

%e ---------------------------------------------------------------

%e 1 {1}

%e 2 6 110 {3, 2}

%e 3 10 1010 {4, 2}

%e 4 18 10010 {5, 2}

%e 5 28 11100 {5, 4, 3}

%e 6 34 100010 {6, 2}

%e 7 44 101100 {6, 4, 3}

%e 8 52 110100 {6, 5, 3}

%e 9 66 1000010 {7, 2}

%e 10 76 1001100 {7, 4, 3}

%e 11 84 1010100 {7, 5, 3}

%e 12 100 1100100 {7, 6, 3}

%e 13 120 1111000 {7, 6, 5, 4}

%e ...

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

%Y Subsequence of A373345.

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

%K nonn,tabf,base,easy

%O 1,2

%A _Paolo Xausa_, Jun 09 2024