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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

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

a(n) <= A131529(n).

Consider the square array in the Example section.

From the second row on, every term t in the array is the difference between the two integers a and b above it (a is the one immediately above t and b is the one to the right of a). There are two ways to compute this term t: t = a - b or t = b - a. Here we always try first to compute t as "the smallest term minus the biggest term". If this operation produces at some point in the antidiagonal a term already present in the array, we stop to compute the successive differences and try instead "the biggest term minus the smallest term". If this operation fails too (we obtain at some point a term already in the array), we have to prolong the first row with another term k, this term being always the smallest available one not present in the array and not leading to a contradiction at any stage in the antidiagonal that k towers.

First row is A175498. Main diagonal is A327743.

The index of where the m-th row first differs from A327743 is 6, 15, 15, 16, 16, 194, 301, 301, 1036, 1036, 1036, 1037, ...

For example, T(6, 194) != A327743(194), but T(6, n) = A327743(n) for n < 194.

Absolute differences of order 1 means k1 = {abs(a(0)-a(1)), abs(a(1)-a(2)), ...} and of order 2: k2 = {abs(k1(0)-k1(1)), abs(k1(1)-k1(2)), ...}.

If we did not allow any duplicated values in k1, k2, ..., kn we would get a case where all k1 .. kn would be the same sequence and this sequence would be s(n) = 2^n. In the case of this sequence k1..kn are not equal but there is still a remarkably strong correlation between a(n), k1(n) and kn(n).

It appears that if a(n) = p then the greatest possible number a(n-1) would be 1 + p + 2^p. If true this would have the consequence that this sequence would be a permutation of the positive integers, because in the case of a(n) = p, n could then not be greater than p + 2^p.

It appears that the logarithmic plot of this sequence consists of straight-line segments attached to each other; this indicates intervals of exponential growth.

If this sequence is a permutation of positive integers, will all k1 .. kn then contain all positive integers at least once?

A subsequence must be of length 2 or greater.