A236335 -id:A236335 - 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

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);

A236336 Lexicographically earliest increasing sequence of positive integers whose graph has no three collinear points.

Original entry on oeis.org

1, 2, 4, 5, 9, 12, 16, 22, 26, 33, 38, 45, 53, 60, 61, 76, 86, 91, 92, 97, 111, 112, 121, 134, 135, 147, 148, 150, 153, 157, 167, 180, 200, 212, 223, 227, 228, 238, 246, 264, 269, 282, 286, 305, 312, 313, 321, 322, 327, 328, 360, 374, 389, 393, 395, 420, 421

Offset: 1

Tanya Khovanova, Jan 22 2014

nonn

An increasing version of A236335.

			Consider a(5). The previous terms are 1,2,4,5. The value of a(5) can't be 6 because points (3,4),(4,5),(5,6) (corresponding to values a(3),a(4),a(5)) are on the same line: y=x+1. Points (1,1),(3,4),(5,7) are on the same line y=3x/2-1/2, so a(5) can't be 7. Points (2,2),(3,4),(5,8) are on the same line: y=2x-2, so a(5) can't be 8. Thus a(5)=5.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A229037, A185256, A236335.

a:= proc(n) option remember; local i, j, k, ok;
      if n<3 then n
    else for k from 1+a(n-1) do ok:=true;
           for j from n-1 to 2 by -1 while ok do
             for i from j-1 to 1 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
      fi
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 23 2014

g[1] = 1;
g[n_] := g[n] =
  Min[Complement[Range[g[n - 1] + 1, 500],
    Select[Flatten[
      Table[g[k] + (n - k) (g[j] - g[k])/(j - k), {k, n - 2}, {j,
        k + 1, n - 1}]], IntegerQ[#] &]]]
Table[g[k], {k, 50}]

A286091 Lexicographically earliest sequence of positive integers such that the slope between any two points (i, a(i)) and (j, a(j)) is distinct.

Original entry on oeis.org

1, 1, 2, 5, 11, 4, 3, 18, 26, 35, 48, 15, 76, 64, 97, 135, 29, 6, 175, 98, 212, 240, 260, 73, 22, 316, 41, 232, 7, 165, 424, 472, 399, 519, 214, 353, 606, 27, 660, 100, 787, 845, 924, 963, 376, 156, 1095, 766, 356, 621, 1206, 32, 501, 1292, 1409, 1169, 1464

Offset: 1

Peter Kagey, May 01 2017

nonn

a(1) = a(2) = 1 appears twice; it is the only term that can appear more than once.
From Peter Kagey, May 02 2017: (Start)
Sequence is bounded above by (n-1)*A000217(n-2)+1.
Proof: The first n terms of the sequences have (1+2+...+(n-1)) = A000217(n-2) slopes, thus all of the lines starting at any of the first (n - 1) points with any of the already-present slopes can at most cross (n, 1), (n, 2), ... (n, (n-1)*A000217(n-2)). (End)

			a(3) != 1 otherwise the slope(a(1),a(2)) = slope(a(1),a(3)), therefore
a(3) = 2.
a(4) != 1 otherwise the slope(a(1),a(2)) = slope(a(1),a(4)),
a(4) != 2 otherwise the slope(a(1),a(2)) = slope(a(3),a(4)),
a(4) != 3 otherwise the slope(a(2),a(3)) = slope(a(3),a(4)),
a(4) != 4 otherwise the slope(a(2),a(3)) = slope(a(1),a(4)), therefore
a(4) = 5.

Peter Kagey and David A. Corneth, Table of n, a(n) for n = 1..1000 (first 600 terms from Peter Kagey)
David A. Corneth, PARI program

Cf. A236335.

A[1]:= 1:
Slopes:= {}:
for n from 2 to 100 do
  for k from 1 do
    Sk:= {seq((k-A[i])/(n-i),i=1..n-1)};
    if Sk intersect Slopes = {} then
      A[n]:= k; Slopes:= Slopes union Sk; break
    fi
od od:
seq(A[n],n=1..100); # Robert Israel, May 01 2017

\\ See link "PARI program". David A. Corneth, May 05 2017

A363554 a(1) = 1; for n > 1, a(n) is the smallest positive integer such that both the gradients and y-intercepts of the lines between any two points (i, a(i)) and (j, a(j)) are distinct.

Original entry on oeis.org

1, 1, 2, 5, 11, 4, 3, 18, 26, 35, 48, 66, 16, 99, 129, 27, 67, 149, 190, 8, 235, 259, 285, 348, 276, 34, 24, 97, 362, 170, 155, 15, 504, 464, 9, 639, 449, 173, 391, 768, 577, 682, 836, 937, 598, 438, 94, 6, 1063, 1007, 500, 210, 1146, 1303, 1390, 806, 1530, 62, 1096, 1739, 212, 28, 1001, 1380

Offset: 1

Scott R. Shannon, Jun 10 2023

nonn

This is a variation of A286091 where the y-intercepts of all lines are also distinct.

			a(12) = 66. A value of 15, with coordinate (12,15), for this term would create a point for which all line gradients are distinct, see A286091, but it creates a line that passes through the origin with a(4), a point with coordinate (4,5). However the terms a(3), at coordinate (3,2) and a(6), at coordinate (6,4), have already created a line that passes through the origin, thus a(12) cannot be 15. The coordinate (12,66) is the first point the leads to all lines and y-intercepts being distinct.

Scott R. Shannon, Table of n, a(n) for n = 1..600

Cf. A286091, A236335, A229037.

A305921 Lexicographically earliest sequence of nonnegative integers such that no three points (i,a(i)), (j,a(j)), (n,a(n)) are collinear and no four points (i,a(i)), (j,a(j)), (k,a(k)), (n,a(n)) are on a circle.

Original entry on oeis.org

0, 0, 1, 1, 5, 3, 8, 2, 3, 2, 4, 8, 9, 7, 15, 11, 4, 10, 5, 11, 16, 9, 30, 38, 26, 30, 18, 10, 28, 36, 17, 21, 38, 7, 12, 20, 49, 41, 23, 23, 6, 16, 28, 13, 6, 29, 49, 56, 17, 19, 36, 22, 24, 56, 64, 12, 61, 21, 14, 69, 13, 68, 78, 53, 33, 69, 39, 27, 31, 18

Offset: 1

Luca Petrone, Jun 14 2018

nonn

			The sequence starts like A236266, but a(5) cannot be 4, because (1,0), (2,0), (4,1) and (5,4) lie on the same circle.

Cf. A236266, A236335.

a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = Module[{i, j, k, l, AB, AC, CD, BC, BD, AD, ok, ok1}, For[l = 0, True, l++, 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)*(l - a[j])]]; If[ok, For[k = n - 1, ok && k >= 1, k--, For[j = k - 1, ok && j >= 0, j--, For[i = j - 1, ok && i >= 0, i--, AB = ((a[i] - a[j])^2 + (i - j)^2)^0.5; AC = ((a[i] - a[k])^2 + (i - k)^2)^0.5; CD = ((a[k] - l)^2 + (k - n)^2)^0.5; BC = ((a[k] - a[j])^2 + (k - j)^2)^0.5; BD = ((a[j] - l)^2 + (j - n)^2)^0.5; AD = ((a[i] - l)^2 + (i - n)^2)^0.5; ok = AB*CD + BC*AD != AC*BD;];];]; If[ok, Return[l]]]]] (* Luca Petrone Jun 17 2018, based on A236266 program by Jean-François Alcover *)

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A231334 Lexicographically earliest sequence of distinct positive integers such that for any distinct i,j, k, the points at positions (i, a(i)), (j, a(j)), (k, a(k)) are not aligned.

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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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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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A236336 Lexicographically earliest increasing sequence of positive integers whose graph has no three collinear points.

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A286091 Lexicographically earliest sequence of positive integers such that the slope between any two points (i, a(i)) and (j, a(j)) is distinct.

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A363554 a(1) = 1; for n > 1, a(n) is the smallest positive integer such that both the gradients and y-intercepts of the lines between any two points (i, a(i)) and (j, a(j)) are distinct.

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