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

1,2

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

Links

Table of n, a(n) for n=1..33.

Rok Cestnik, Graphical example

David A. Corneth, Seqfan post about this sequence, May 01 2017.

Example

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

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 *)

Crossrefs

Cf. A231334, A236335, A301802.

Sequence in context: A366263 A231334 A253609 * A082560 A191598 A338221

Adjacent sequences: A299999 A300000 A300001 * A300003 A300004 A300005

Keyword

nonn,hard,more,nice,changed

Author

Peter Kagey, Apr 17 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

Status

approved