A300002 Lexicographically earliest sequence of positive integers such that no k+2 points fall on any polynomial of degree k.

1, 2, 4, 3, 6, 5, 9, 16, 14, 20, 7, 15, 8, 12, 18, 31, 26, 27, 40, 30, 49, 38, 19, 10, 23, 53, 11, 32, 21, 25, 13, 47, 83

Offset: 1

Author

Peter Kagey, Apr 17 2017

Keywords

• nonn
• hard
• more
• nice

Comments

a(n) must avoid 2^(n-1)-1 polynomials: the polynomials defined by each nonempty subset of the first (n-1) terms of the sequence.

Conjecture: This sequence is a permutation of the natural numbers.

From David A. Corneth, May 10 2017: (Start)

Sequence is also "Lexicographically earliest sequence of positive integers such that any k+1 points fall on a polynomial of degree k."

Conjecture: a(27)-a(32) are 11, 32, 21, 25, 13, 47. If all previous data are correct, no polynomial of degree ceiling(n/2.5) - 1 goes through any set of points. (End)

Formerly A285175. - Peter Kagey, Mar 06 2018

Examples

			a(1) = 1.
a(2) != 1 or else (1, 1) and (2, 1) fall on y = 1. (Similarly all terms must be distinct.)
a(2) = 2.
a(3) != 1 or else (1, 1) and (3, 1) fall on y = 1.
a(3) != 2 or else (2, 2) and (3, 2) fall on y = 2.
a(3) != 3 or else (1, 1), (2, 2) and (3, 3) fall on y = x.
a(3) = 4.
a(4) != 1 or else (1, 1) and (4, 1) fall on y = 1.
a(4) != 2 or else (2, 2) and (4, 2) fall on y = 2.
a(4) = 3
		

Links

• Rok Cestnik, Graphical example
• David A. Corneth, Seqfan post about this sequence, May 01 2017.

Crossrefs

Cf. A231334, A236335, A301802.

Programs

Mathematica

A = {{1, 1}, {2, 2}};
n = 3;
While[n < 50,
c = Sort[Select[Select[InterpolatingPolynomial[#, n] & /@ Subsets[A, {1, n - 1}], # > 0 & ] , IntegerQ]];
B = Differences[c];
If[Max[B] == 1,
d = Max[c] + 1,
d = Part[c, First[Position[B, Select[B, # > 1 &][[1]]]][[1]]] + 1];
A = Append[A, {n, d}];
Print[{n, d}]
n++;
] (* Luca Petrone, Apr 18 2017 *)

Extensions

a(21)-a(26) from Luca Petrone, Apr 19 2017

a(27) from Robert G. Wilson v, Jul 09 2017

a(28)-a(33) from Bert Dobbelaere, Apr 12 2024