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A327743
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a(n) = smallest positive number not already in the sequence such that for each k = 1, ..., n-1, the k-th differences are distinct.
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9
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1, 2, 4, 3, 6, 11, 5, 9, 7, 13, 10, 18, 8, 15, 27, 14, 23, 12, 22, 17, 28, 16, 29, 20, 34, 19, 35, 21, 36, 32, 24, 42, 26, 43, 25, 44, 66, 33, 53, 30, 51, 31, 54, 37, 61, 39, 64, 38, 67, 40, 70, 41, 68, 47, 75, 50, 76, 45, 77, 49, 80, 48, 81, 46, 82, 52, 86
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OFFSET
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1,2
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COMMENTS
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Is this sequence a permutation of the positive integers?
Does each k-th difference contain all nonzero integers?
It is not difficult to show that if a(1), ..., a(k) satisfy the requirements, then any sufficiently large number is a candidate for a(k+1). So a(k) exists for all k. - N. J. A. Sloane, Sep 24 2019
The original definition was "Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, the k-th differences are distinct."
If only first differences are considered, one gets the classical Mian-Chowla sequence A005282. - M. F. Hasler, Oct 09 2019
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LINKS
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EXAMPLE
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Illustration of the first eight terms of the sequence.
k | k-th differences
--+---------------------------------
0 | 1, 2, 4, 3, 6, 11, 5, 9
1 | 1, 2, -1, 3, 5, -6, 4
2 | 1, -3, 4, 2, -11, 10
3 | -4, 7, -2, -13, 21
4 | 11, -9, -11, 34
5 | -20, -2, 45
6 | 18, 47
7 | 29
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MATHEMATICA
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a[1] = 1;
a[n_] := a[n] = For[aa = Array[a, n-1]; an = 1, True, an++, If[FreeQ[aa, an], aa = Append[aa, an]; If[AllTrue[Range[n-1], Unequal @@ Differences[ aa, #]&], Return[an]]]];
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CROSSREFS
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First differences: A327452; leading column of difference triangle: A327457.
If ALL terms of the difference triangle must be distinct, see A327460 and A327762.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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"Infinite" added to definition (for otherwise the one-term sequence 1 is earlier). - N. J. A. Sloane, Sep 25 2019
Changed definition to avoid use of "Lexicographically earliest infinite sequence" and the associated existence questions. - N. J. A. Sloane, Sep 28 2019
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STATUS
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approved
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