A373556 - OEIS

%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