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A368400
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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.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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Triangle begins:
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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):
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100 101 | 4 5 | 1 2
010 110 | 2 6 | 3 4
000 001 011 111 | 0 1 3 7 | 5 6 7 8
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The position of the elements within the pattern is therefore the following:
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Element: 0 1 2 3 4 5 6 7
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V V V V V V V V
Position: 5 6 3 7 1 2 4 8
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MATHEMATICA
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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]]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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