cp's OEIS Frontend

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

%I A300002 #28 Dec 23 2024 14:53:45
%S A300002 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 A300002 11,32,21,25,13,47,83
%N A300002 Lexicographically earliest sequence of positive integers such that no k+2 points fall on any polynomial of degree k.
%C A300002 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 A300002 Conjecture: This sequence is a permutation of the natural numbers.
%C A300002 From _David A. Corneth_, May 10 2017: (Start)
%C A300002 Sequence is also "Lexicographically earliest sequence of positive integers such that any k+1 points fall on a polynomial of degree k."
%C A300002 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 A300002 Formerly A285175. - _Peter Kagey_, Mar 06 2018
%H A300002 Rok Cestnik, <a href="/A300002/a300002.pdf">Graphical example</a>
%H A300002 David A. Corneth, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-May/017517.html">Seqfan post about this sequence</a>, May 01 2017.
%e A300002 a(1) = 1.
%e A300002 a(2) != 1 or else (1, 1) and (2, 1) fall on y = 1. (Similarly all terms must be distinct.)
%e A300002 a(2) = 2.
%e A300002 a(3) != 1 or else (1, 1) and (3, 1) fall on y = 1.
%e A300002 a(3) != 2 or else (2, 2) and (3, 2) fall on y = 2.
%e A300002 a(3) != 3 or else (1, 1), (2, 2) and (3, 3) fall on y = x.
%e A300002 a(3) = 4.
%e A300002 a(4) != 1 or else (1, 1) and (4, 1) fall on y = 1.
%e A300002 a(4) != 2 or else (2, 2) and (4, 2) fall on y = 2.
%e A300002 a(4) = 3
%t A300002 A = {{1, 1}, {2, 2}};
%t A300002 n = 3;
%t A300002 While[n < 50,
%t A300002 c = Sort[Select[Select[InterpolatingPolynomial[#, n] & /@ Subsets[A, {1, n - 1}], # > 0 & ] , IntegerQ]];
%t A300002 B = Differences[c];
%t A300002 If[Max[B] == 1,
%t A300002 d = Max[c] + 1,
%t A300002 d = Part[c, First[Position[B, Select[B, # > 1 &][[1]]]][[1]]] + 1];
%t A300002 A = Append[A, {n, d}];
%t A300002 Print[{n, d}]
%t A300002 n++;
%t A300002 ] (* _Luca Petrone_, Apr 18 2017 *)
%Y A300002 Cf. A231334, A236335, A301802.
%K A300002 nonn,hard,more,nice
%O A300002 1,2
%A A300002 _Peter Kagey_, Apr 17 2017
%E A300002 a(21)-a(26) from _Luca Petrone_, Apr 19 2017
%E A300002 a(27) from _Robert G. Wilson v_, Jul 09 2017
%E A300002 a(28)-a(33) from _Bert Dobbelaere_, Apr 12 2024