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
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.
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
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..20000
Programs
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Maple
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);
Comments
Examples
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.Links
Crossrefs
Programs
Maple
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);Mathematica
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 *)Formula