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A213222 Minimum number of distinct slopes formed by n noncollinear points in the plane.

%I A213222 #36 Feb 16 2025 08:33:17
%S A213222 3,4,4,6,6,8,8,10,10,12,12,14,14,16,16,18,18,20,20,22,22,24,24,26,26,
%T A213222 28,28,30,30,32,32,34,34,36,36,38,38,40,40,42,42,44,44,46,46,48,48,50,
%U A213222 50,52,52,54,54,56,56,58,58,60,60,62,62,64,64,66,66,68,68,70,70,72,72,74,74,76,76,78,78,80,80,82,82,84
%N A213222 Minimum number of distinct slopes formed by n noncollinear points in the plane.
%C A213222 Scott formulated the problem (on the basis of a similar problem of Erdős), gave bounds, and conjectured the formula which Unger later proved.
%C A213222 Also the edge chromatic number of the n-polygon diagonal intersection graph. - _Eric W. Weisstein_, Mar 23 2018
%D A213222 Martin Aigner and Gunter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin, 2000. Chapter 10.
%H A213222 Vincenzo Librandi, <a href="/A213222/b213222.txt">Table of n, a(n) for n = 3..1000</a>
%H A213222 J. E. Goodman and R. Pollack, <a href="http://dx.doi.org/10.1016/0097-3165(80)90011-4">On the combinatorial classification of nondegenerate configurations in the plane</a>, J. Combin. Theory Ser. A, 29 (1980), pp. 220-235.
%H A213222 Giovanni Resta, <a href="/A213222/a213222.png">Illustration of terms a(3)-a(7)</a>
%H A213222 P. R. Scott, <a href="http://www.jstor.org/stable/2317384">On the sets of directions determined by n points</a>, The American Mathematical Monthly 77:5 (1970), pp. 502-505.
%H A213222 Peter Ungar, <a href="http://dx.doi.org/10.1016/0097-3165(82)90045-0">2N noncollinear points determine at least 2N directions</a>, Journal of Combinatorial Theory, Series A, 33:3 (1982), pp. 343-347.
%H A213222 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeChromaticNumber.html">Edge Chromatic Number</a>
%H A213222 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html">Polygon Diagonal Intersection Graph</a>
%H A213222 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A213222 a(n) = 2*floor(n/2) for n > 3.
%F A213222 G.f.: x^3*(3+x-3*x^2+x^3)/((1+x)*(1-x)^2). [_Bruno Berselli_, Mar 04 2013]
%p A213222 A213222:=n->`if`(n = 3, 3, 2*floor(n/2)); seq(A213222(n), n=3..100); # _Wesley Ivan Hurt_, Mar 28 2014
%t A213222 CoefficientList[Series[(3 + x - 3 x^2 + x^3)/((1 + x) (1 - x)^2), {x, 0, 100}], x] (* _Vincenzo Librandi_, Mar 29 2014 *)
%t A213222 LinearRecurrence[{1,1,-1},{3,4,4,6},100] (* _Harvey P. Dale_, Dec 29 2024 *)
%o A213222 (PARI) a(n)=if(n>3,n\2*2,3)
%o A213222 (Magma) [2*Floor(n/2): n in [3..100]]; // _Vincenzo Librandi_, Mar 29 2014
%Y A213222 Cf. A000769, A000755, A219760.
%Y A213222 Cf. A000217 (maximum number of slopes, with offset 1).
%K A213222 nonn,easy
%O A213222 3,1
%A A213222 _Charles R Greathouse IV_, Mar 02 2013