A368399 Irregular triangle read by rows: row n lists the indices of rows of the Christmas tree pattern (A367508) of order n, sorted by row length and, in case of ties, by row index.

1, 1, 2, 1, 2, 3, 1, 3, 2, 4, 5, 6, 1, 2, 4, 5, 7, 3, 6, 8, 9, 10, 1, 3, 7, 9, 13, 2, 4, 5, 8, 10, 11, 14, 15, 17, 6, 12, 16, 18, 19, 20, 1, 2, 4, 5, 7, 11, 12, 14, 15, 17, 21, 22, 24, 28, 3, 6, 8, 9, 13, 16, 18, 19, 23, 25, 26, 29, 30, 32, 10, 20, 27, 31, 33, 34, 35

Offset

1,3

Comments

Row n is a permutation of the integers in the interval [1, binomial(n,floor(n/2))].

See A367508 for the description of the Christmas tree patterns, references and links.

Links

Paolo Xausa, Table of n, a(n) for n = 1..13494 (rows 1..15 of the triangle, flattened).

Example

Triangle begins (vertical bars separate indices of rows having different lengths):
.
  [1]  1;
  [2]  1| 2;
  [3]  1  2| 3;
  [4]  1  3| 2  4  5| 6;
  [5]  1  2  4  5  7| 3  6  8  9|10;
  [6]  1  3  7  9 13| 2  4  5  8 10 11 14 15 17| 6 12 16 18 19|20;
  ...
For example, the order 4 of the Christmas tree pattern is the following:
.
            1010             Row 1 length = 1
       1000 1001 1011        Row 2 length = 3
            1100             Row 3 length = 1
       0100 0101 1101        Row 4 length = 3
       0010 0110 1110        Row 5 length = 3
  0000 0001 0011 0111 1111   Row 6 length = 5
.
and ordering the rows by length (and then by row index) gives 1, 3, 2, 4, 5, 6.

Mathematica

With[{nmax=8}, Map[Flatten[Values[PositionIndex[#]]]&, SubstitutionSystem[{1->{2}, t_/; t>1:>{t-1, t+1}}, {2}, nmax-1]]]

Crossrefs

Cf. A001405, A363718 (row lengths), A367508, A368400.

Sequence in context: A370408 A358090 A286343 * A075106 A196935 A226314

Adjacent sequences: A368396 A368397 A368398 * A368400 A368401 A368402

Keyword

nonn,tabf

Author

Paolo Xausa, Dec 23 2023

Status

approved