A300002 - OEIS

%I #27 Apr 12 2024 07:22:53

%S 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,

%T 11,32,21,25,13,47,83

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

%C 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.

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

%C From _David A. Corneth_, May 10 2017: (Start)

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

%C 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)

%C Formerly A285175. - _Peter Kagey_, Mar 06 2018

%H Rok Cestnik, <a href="/A300002/a300002.pdf">Graphical example</a>

%H David A. Corneth, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-May/017517.html">Seqfan post about this sequence</a>, May 01 2017.

%e a(1) = 1.

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

%e a(2) = 2.

%e a(3) != 1 or else (1, 1) and (3, 1) fall on y = 1.

%e a(3) != 2 or else (2, 2) and (3, 2) fall on y = 2.

%e a(3) != 3 or else (1, 1), (2, 2) and (3, 3) fall on y = x.

%e a(3) = 4.

%e a(4) != 1 or else (1, 1) and (4, 1) fall on y = 1.

%e a(4) != 2 or else (2, 2) and (4, 2) fall on y = 2.

%e a(4) = 3

%t A = {{1, 1}, {2, 2}};

%t n = 3;

%t While[n < 50,

%t c = Sort[Select[Select[InterpolatingPolynomial[#, n] & /@ Subsets[A, {1, n - 1}], # > 0 & ] , IntegerQ]];

%t B = Differences[c];

%t If[Max[B] == 1,

%t d = Max[c] + 1,

%t d = Part[c, First[Position[B, Select[B, # > 1 &][[1]]]][[1]]] + 1];

%t A = Append[A, {n, d}];

%t Print[{n, d}]

%t n++;

%t ] (* _Luca Petrone_, Apr 18 2017 *)

%Y Cf. A231334, A236335, A301802.

%K nonn,hard,more,nice

%O 1,2

%A _Peter Kagey_, Apr 17 2017

%E a(21)-a(26) from _Luca Petrone_, Apr 19 2017

%E a(27) from _Robert G. Wilson v_, Jul 09 2017

%E a(28)-a(33) from _Bert Dobbelaere_, Apr 12 2024