A327459 -id:A327459 - 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.

A327762 a(n) = smallest positive number not already in the sequence such that all n(n+1)/2 numbers in the triangle of differences of the first n terms are distinct.

Original entry on oeis.org

Offset: 1

N. J. A. Sloane, Sep 24 2019, revised Sep 25 2019

Inspired by A327743.
From Rémy Sigrist, Sep 25 2019: (Start)
The sequence is finite, with 56 terms.
Let b and c be the first and second differences of a, respectively, hence:
- b(55) = a(56) - a(55) = 99 - 198 = -99,
- b(56) = a(57) - a(56) = a(57) - 99,
- c(55) = b(56) - b(55) = a(57), a contradiction.
(End)
Since this definition leads to a finite sequence, it is natural to ask instead for the "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." This is A327460.
If only first differences are considered, one gets the classical Mian-Chowla sequence A005282. - M. F. Hasler, Oct 09 2019

			Difference triangle of the first k=8 terms of the sequence:
  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.

Cf. A327743, A327460.
For first differences see A327458; for the leading column of the difference triangle see A327459.
Cf. A005282.

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.

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