A231334 -id:A231334 - cp's OEIS frontend

A236266 Lexicographically earliest sequence of nonnegative integers such that no three points (i,a(i)), (j,a(j)), (n,a(n)) are collinear.

0, 0, 1, 1, 4, 3, 8, 2, 2, 5, 7, 4, 5, 8, 16, 3, 7, 14, 12, 23, 16, 12, 25, 31, 13, 6, 11, 28, 11, 17, 9, 9, 22, 34, 6, 15, 13, 29, 23, 22, 29, 45, 26, 19, 51, 14, 24, 39, 28, 39, 18, 37, 57, 17, 38, 41, 15, 68, 32, 24, 66, 42, 10, 50, 27, 10, 53, 72, 25, 26

Offset: 0

Alois P. Heinz, Jan 21 2014

(a(n)-a(j))/(n-j) <> (a(j)-a(i))/(j-i) for all 0<=i

			For n=4 the value of a(n) cannot be less than 4 because otherwise we would have a set of three collinear points, {(0,0),(1,0),(4,0)} or {(2,1),(3,1),(4,1)} or {(0,0),(2,1),(4,2)} or {(1,0),(2,1),(4,3)}.  Thus a(4) = 4 is the first value that is in accordance with the constraints.

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Dániel T. Nagy, Zoltán Lóránt Nagy, and Russ Woodroofe, The extensible No-Three-In-Line problem, arXiv:2209.01447 [math.CO], 2022.

Cf. A005836, A179040, A231334, A236335, A255708, A255709.

a:= proc(n) option remember; local i, j, k, ok;
      for k from 0 do ok:=true;
        for j from n-1 to 1 by -1 while ok do
          for i from j-1 to 0 by -1 while ok do
            ok:= (n-j)*(a(j)-a(i))<>(j-i)*(k-a(j)) od
        od; if ok then return k fi
      od
    end:
seq(a(n), n=0..60);

a[0] = a[1] = 0; a[n_] := a[n] = Module[{i, j, k, ok}, For[k = 0, True, k++, ok = True; For[j = n-1, ok && j >= 1, j--, For[i = j-1, ok && i >= 0, i--, ok = (n-j)*(a[j]-a[i]) != (j-i)*(k-a[j])]]; If[ok, Return[k]]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Jun 16 2018, after Alois P. Heinz *)

a(n) = A236335(n+1) - 1. - Alois P. Heinz, Jan 23 2014

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

Original entry on oeis.org

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

Peter Kagey, Apr 17 2017

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

			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

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

Cf. A231334, A236335, A301802.

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 *)

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

A334043 a(1) = 0, and for any n > 1, a(n) is the number of points of the set { (k, a(k)), k = 1..n-2 } that are visible from the point (n-1, a(n-1)).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 5, 4, 5, 7, 8, 8, 10, 8, 9, 12, 11, 13, 16, 14, 15, 16, 14, 17, 20, 20, 17, 21, 25, 23, 26, 28, 27, 25, 29, 25, 31, 27, 34, 34, 28, 39, 35, 36, 41, 36, 40, 41, 41, 42, 45, 35, 49, 45, 47, 46, 49, 47, 49, 47, 54, 54, 52, 56, 54, 54, 58, 56, 59

Offset: 1

Rémy Sigrist, Apr 13 2020

nonn

For any i and k such that i < k: the point (i, a(i)) is visible from the point (k, a(k)) if there are no j such that i < j < k and the three points (i, a(i)), (j, a(j)), (k, a(k)) are aligned.

			For n = 5:
- we consider the following points:
        .   .   .   X
                  /  (4,2)
        .   .   X   .
              /  (3,1)
        X   X   .   .
   (1,0)     (2,0)
- (1,0) and (3,1) are visible from (4,2)
- whereas (2,0) is not visible from (4,2),
- hence a(5) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

See A334044 for a similar sequence.

Cf. A231334.

g(z) = z/gcd(real(z), imag(z))
for (n=1, #a=vector(69), print1 (a[n] = #Set(apply(k -> g((k+a[k]*I)-(n-1+a[n-1]*I)), [1..n-2])) ", "))

A255708 No three points (i,a(i)), (j,a(j)), (k,a(k)) are collinear, for n = 0,1,2,... the value of a(n) is chosen to be m or -m (in this order) for the smallest m>=0 satisfying the condition.

Original entry on oeis.org

0, 0, 1, 1, -1, -1, 4, 2, 2, -3, -5, -2, -7, -2, 5, 3, 3, -5, -4, -4, 6, 5, -6, -3, -10, 11, -6, 4, 18, 11, 19, 7, 12, 12, 6, -13, 19, 7, -10, -7, -9, -14, 13, 23, -28, -8, -14, 9, 8, -22, -9, -8, 23, -11, 15, 22, 13, 8, -21, -13, -26, 9, -12, -12, -11, 40, 21

Offset: 0

Alois P. Heinz, Mar 03 2015

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A005836, A179040, A231334, A236266, A236335, A255709.

a:= proc(n) option remember; local i, j, k, t, ok;
      for t from 0 do for k in [t, -t] do ok:=true;
        for j from n-1 to 1 by -1 while ok do
          for i from j-1 to 0 by -1 while ok do
            ok:= (n-j)*(a(j)-a(i))<>(j-i)*(k-a(j))
        od od; if ok then return k fi
      od od
    end:
seq(a(n), n=0..60);

A255709 No three points (i,a(i)), (j,a(j)), (k,a(k)) are collinear and all values distinct, for n = 0,1,2,... the value of a(n) is chosen to be m or -m (in this order) for the smallest m>=0 satisfying the condition.

Original entry on oeis.org

0, 1, -1, 2, 3, -2, -5, -3, 4, -6, 6, -7, -4, 5, 12, 16, 7, 8, -10, -8, 9, 19, 14, -12, -14, -9, 21, 10, -11, -15, 17, 15, -19, 13, -22, -13, -16, -24, 11, 18, 22, -18, 25, 23, -17, 24, 40, -21, -38, 20, -29, 36, -30, -20, 32, -34, 26, 43, -23, 37, -26, 33

Offset: 0

Alois P. Heinz, Mar 03 2015

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A005836, A179040, A231334, A236266, A236335, A255708.

b:= proc() true end:
a:= proc(n) option remember; local i, j, k, t, ok;
      for t from 0 do for k in [t, -t] do ok:=b(k);
        for j from n-1 to 1 by -1 while ok do
          for i from j-1 to 0 by -1 while ok do
            ok:= (n-j)*(a(j)-a(i))<>(j-i)*(k-a(j))
        od od; if ok then b(k):=false; return k fi
      od od
    end:
seq(a(n), n=0..60);

A300670 Table read by antidiagonals: the n-th row is the lexicographically earliest sequence such that no k + 2 points of ((1, a(1)), (2, a(2)), ...) lie on a polynomial of degree k for k < n.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 3, 4, 2, 1, 6, 6, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 8, 9, 5, 6, 3, 4, 2, 1, 9, 12, 9, 5, 6, 3, 4, 2, 1, 10, 7, 12, 9, 5, 6, 3, 4, 2, 1, 11, 14, 19, 12, 9, 5, 6, 3, 4, 2, 1, 12, 13, 17, 19, 16, 9, 5, 6, 3, 4, 2, 1, 13, 8, 7, 17

Offset: 1

Peter Kagey, Mar 11 2018

Is every row a permutation of the natural numbers?
The first row is the positive integers, the second row is A231334, and the main diagonal is A300002.
T(n, m) = A300002(m) for n >= m, thus the rows converge to A300002 in the limit.

			Table begins
1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, 16, ...
1, 2, 4, 3, 6, 5, 9, 12,  7, 14, 13,  8, 23, 17, 18, 22, ...
1, 2, 4, 3, 6, 5, 9, 12, 19, 17,  7,  8, 15, 20, 18, 22, ...
1, 2, 4, 3, 6, 5, 9, 12, 19, 17,  8, 10, 31,  7, 11, 22, ...
1, 2, 4, 3, 6, 5, 9, 16, 14, 20,  7, 15,  8, 12, 18, 31, ...
1, 2, 4, 3, 6, 5, 9, 16, 14, 20,  7, 15,  8, 12, 18, 31, ...
...
In the first row, no two points lie on a 0-degree polynomial (i.e., all terms are distinct).
In the second row, no two terms are the same and no three points (1, a(1)), (2, a(2)), ... lie on the same line.
In the third row, no two terms are the same; no three points (1, a(1)), (2, a(2)), ... lie on the same line; and no four points lie on the same parabola.

Cf. A231334, A300002.

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A236335 Lexicographically earliest sequence of positive integers whose graph has no three collinear points.

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A236266 Lexicographically earliest sequence of nonnegative integers such that no three points (i,a(i)), (j,a(j)), (n,a(n)) are collinear.

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

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A334043 a(1) = 0, and for any n > 1, a(n) is the number of points of the set { (k, a(k)), k = 1..n-2 } that are visible from the point (n-1, a(n-1)).

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A255708 No three points (i,a(i)), (j,a(j)), (k,a(k)) are collinear, for n = 0,1,2,... the value of a(n) is chosen to be m or -m (in this order) for the smallest m>=0 satisfying the condition.

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A255709 No three points (i,a(i)), (j,a(j)), (k,a(k)) are collinear and all values distinct, for n = 0,1,2,... the value of a(n) is chosen to be m or -m (in this order) for the smallest m>=0 satisfying the condition.

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Maple

A300670 Table read by antidiagonals: the n-th row is the lexicographically earliest sequence such that no k + 2 points of ((1, a(1)), (2, a(2)), ...) lie on a polynomial of degree k for k < n.

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