

A300002


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


5



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
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OFFSET

1,2


COMMENTS

a(n) must avoid 2^(n1)1 polynomials: the polynomials defined by each nonempty subset of the first (n1) 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..27.
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.
Sequence in context: A232643 A231334 A253609 * A082560 A191598 A338221
Adjacent sequences: A299999 A300000 A300001 * A300003 A300004 A300005


KEYWORD

nonn,more,nice


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


STATUS

approved



