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.

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, 99

Offset: 1

Author

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

Keywords

• nonn
• full
• fini

Comments

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

Examples

			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.
		

Crossrefs

Cf. A327743, A327460.

For first differences see A327458; for the leading column of the difference triangle see A327459.

Cf. A005282.