A368400 Irregular triangle read by rows: T(n,k) is the position of k within the Christmas tree pattern (A367562) of order n, with n >= 1 and k >= 0.

1, 2, 2, 3, 1, 4, 5, 6, 3, 7, 1, 2, 4, 8, 12, 13, 9, 14, 6, 7, 10, 15, 2, 3, 1, 4, 5, 8, 11, 16, 27, 28, 23, 29, 19, 20, 24, 30, 13, 14, 11, 15, 17, 21, 25, 31, 5, 6, 3, 7, 1, 2, 4, 8, 9, 10, 12, 16, 18, 22, 26, 32, 58, 59, 53, 60, 48, 49, 54, 61, 40, 41, 37, 42, 45

Offset: 1

Author

Paolo Xausa, Dec 23 2023

Keywords

• nonn
• base
• tabf
• look

Comments

Row n is a permutation of the integers in the interval [1, 2^n].

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

Examples

			Triangle begins:
.
  n\k|  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  --------------------------------------------------------
   1 |  1  2
   2 |  2  3  1  4
   3 |  5  6  3  7  1  2  4  8
   4 | 12 13  9 14  6  7 10 15  2  3  1  4  5  8 11 16
  ...
For example, the order 3 of the Christmas tree pattern is the following (binary on the left, converted to decimal in the middle, position within the pattern on the right):
.
      100 101     |   4 5   |   1 2
      010 110     |   2 6   |   3 4
  000 001 011 111 | 0 1 3 7 | 5 6 7 8
.
The position of the elements within the pattern is therefore the following:
.
  Element:  0  1  2  3  4  5  6  7
            |  |  |  |  |  |  |  |
            V  V  V  V  V  V  V  V
  Position: 5  6  3  7  1  2  4  8
.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..8190 (rows 1..12 of the triangle, flattened).

Crossrefs

Cf. A367508, A367562, A368399.

Programs

Mathematica

A367562list[imax_]:=Map[FromDigits[#,2]&,NestList[Map[Delete[{If[Length[#]>1,Map[#<>"0"&,Rest[#]],Nothing],Join[{#[[1]]<>"0"},Map[#<>"1"&,#]]},0]&],{{"0","1"}},imax-1],{3}];
With[{nmax=6},Map[Flatten[Values[KeySort[PositionIndex[Flatten[#]]]]]&,A367562list[nmax]]]