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A327743 a(n) = smallest positive number not already in the sequence such that for each k = 1, ..., n-1, the k-th differences are distinct. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Peter Kagey, Table of n, a(n) for n = 1..5000

EXAMPLE

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

MATHEMATICA

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]]]];

a /@ Range[1, 100] (* Jean-Fran├žois Alcover, Oct 26 2019 *)

CROSSREFS

Cf. A175498.

First differences: A327452; leading column of difference triangle: A327457.

If ALL terms of the difference triangle must be distinct, see A327460 and A327762.

Cf. A005282.

Sequence in context: A285493 A064273 A257986 * A232564 A134561 A258046

Adjacent sequences:  A327740 A327741 A327742 * A327744 A327745 A327746

KEYWORD

nonn,nice

AUTHOR

Peter Kagey, Sep 24 2019

EXTENSIONS

"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

STATUS

approved

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Last modified October 22 18:45 EDT 2021. Contains 348175 sequences. (Running on oeis4.)