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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.
7
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
0,5
COMMENTS
(a(n)-a(j))/(n-j) <> (a(j)-a(i))/(j-i) for all 0<=i<j<n. No value occurs more than twice. Each triangle with (distinct) vertices (i,a(i)), (j,a(j)), (n,a(n)) has area larger than zero.
LINKS
Dániel T. Nagy, Zoltán Lóránt Nagy, and Russ Woodroofe, The extensible No-Three-In-Line problem, arXiv:2209.01447 [math.CO], 2022.
FORMULA
a(n) = A236335(n+1) - 1. - Alois P. Heinz, Jan 23 2014
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Alois P. Heinz, Jan 21 2014
STATUS
approved