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.

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

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-François 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 * A378846 A232564 A134561

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