A300670 Table read by antidiagonals: the n-th row is the lexicographically earliest sequence such that no k + 2 points of ((1, a(1)), (2, a(2)), ...) lie on a polynomial of degree k for k < n.

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 3, 4, 2, 1, 6, 6, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 8, 9, 5, 6, 3, 4, 2, 1, 9, 12, 9, 5, 6, 3, 4, 2, 1, 10, 7, 12, 9, 5, 6, 3, 4, 2, 1, 11, 14, 19, 12, 9, 5, 6, 3, 4, 2, 1, 12, 13, 17, 19, 16, 9, 5, 6, 3, 4, 2, 1, 13, 8, 7, 17

Offset: 1

Author

Peter Kagey, Mar 11 2018

Keywords

• nonn
• tabl

Comments

Is every row a permutation of the natural numbers?

The first row is the positive integers, the second row is A231334, and the main diagonal is A300002.

T(n, m) = A300002(m) for n >= m, thus the rows converge to A300002 in the limit.

Examples

			Table begins
1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, 16, ...
1, 2, 4, 3, 6, 5, 9, 12,  7, 14, 13,  8, 23, 17, 18, 22, ...
1, 2, 4, 3, 6, 5, 9, 12, 19, 17,  7,  8, 15, 20, 18, 22, ...
1, 2, 4, 3, 6, 5, 9, 12, 19, 17,  8, 10, 31,  7, 11, 22, ...
1, 2, 4, 3, 6, 5, 9, 16, 14, 20,  7, 15,  8, 12, 18, 31, ...
1, 2, 4, 3, 6, 5, 9, 16, 14, 20,  7, 15,  8, 12, 18, 31, ...
...
In the first row, no two points lie on a 0-degree polynomial (i.e., all terms are distinct).
In the second row, no two terms are the same and no three points (1, a(1)), (2, a(2)), ... lie on the same line.
In the third row, no two terms are the same; no three points (1, a(1)), (2, a(2)), ... lie on the same line; and no four points lie on the same parabola.
		

Crossrefs

Cf. A231334, A300002.