A327762 -id:A327762 - cp's OEIS frontend

A327460 Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, all the k(k+1)/2 numbers in the triangle of differences of the first k terms are distinct.

1, 3, 9, 5, 12, 10, 23, 8, 22, 17, 42, 16, 43, 20, 38, 26, 45, 32, 65, 28, 64, 39, 76, 34, 81, 48, 98, 40, 92, 54, 109, 60, 116, 51, 114, 58, 117, 70, 136, 67, 135, 71, 145, 72, 147, 69, 146, 80, 164, 87, 166, 82, 170, 108, 198, 101

Offset: 1

N. J. A. Sloane, Sep 25 2019

This is an infinite version of A327762. The first 55 terms are the same as in A327762.
Inspired by A327743.
The usual topological arguments show that there IS a sequence satisfying the definition. So far, the terms of A327460 lie on two roughly straight lines, of slopes about 1.75 and 3.5: see A328069, A328070. - N. J. A. Sloane, Oct 07 2019
If only the first differences are constrained, one gets the classical Mian-Chowla sequence A005282. - M. F. Hasler, Oct 09 2019. See also another classic, A005228, and A328190. - N. J. A. Sloane, Nov 01 2019

			The difference triangle of the first k=8 terms of the sequence is
     1,    3,    9,   5,  12,  10,  23, 8, ...
     2,    6,   -4,   7,  -2,  13, -15, ...
     4,  -10,   11,  -9,  15, -28, ...
   -14,   21,  -20,  24, -43, ...
    35,  -41,   44, -67, ...
   -76,   85, -111, ...
   161, -196, ...
  -357, ...
All 8*9/2 = 36 numbers are distinct.

Peter Kagey, Table of n, a(n) for n = 1..4000

Cf. A005228, A005282, A035313, A327743, A327762, A328190.
See also A327458 (differences), A328066 (sorted), A328067, A328068 (complement), A328069 and A328070 (bisections), A328071; A235538 (absolute differences distinct).
The inverse binomial transform is A327459.

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.

Original entry on oeis.org

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

Peter Kagey, Sep 24 2019

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

			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

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

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.

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 *)

"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

A327458 First differences of A327460.

Original entry on oeis.org

2, 6, -4, 7, -2, 13, -15, 14, -5, 25, -26, 27, -23, 18, -12, 19, -13, 33, -37, 36, -25, 37, -42, 47, -33, 50, -58, 52, -38, 55, -49, 56, -65, 63, -56, 59, -47, 66, -69, 68, -64, 74, -73, 75, -78, 77, -66, 84, -77, 79, -84, 88, -62, 90, -97, 94, -109, 96, -87, 102, -91, 119, -122

Offset: 1

N. J. A. Sloane, Sep 24 2019

sign

This was originally the first differences of A327762, a finite sequence. Replaced by first differences of A327460, an infinite and more interesting sequence (the first 55 differences are the same).

N. J. A. Sloane, Table of n, a(n) for n = 1..3999

Cf. A327460, A327762.

Entry revised by N. J. A. Sloane, Oct 05 2019

A327459 Inverse binomial transform of A327460.

Original entry on oeis.org

1, 2, 4, -14, 35, -76, 161, -357, 831, -1955, 4508, -10105, 22168, -48146, 104484, -227391, 495414, -1075389, 2313782, -4915216, 10288268, -21214952, 43157269, -86845138, 173498199, -345556983, 689067113, -1380830712, 2788197482, -5680398553, 11675197082, -24179658910, 50364058015, -105288240165, 220499927388

Offset: 0

N. J. A. Sloane, Sep 24 2019

sign

For the first 55 terms or so, this is also the inverse binomial transform of the finite sequence A327762.

N. J. A. Sloane, Table of n, a(n) for n = 0..999

Cf. A327460, A327762.

This was formerly the inverse binomial transform of the finite sequence A327762, but it has been changed to the inverse binomial transform of the infinite sequence A327460. This only affects terms from about term 56 onward. It also makes the sequence infinite. - N. J. A. Sloane, Oct 07 2019

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A327460 Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, all the k(k+1)/2 numbers in the triangle of differences of the first k terms are distinct.

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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.

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A327458 First differences of A327460.

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A327459 Inverse binomial transform of A327460.

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