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

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, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 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

Offset

3,1

Comments

Scott formulated the problem (on the basis of a similar problem of Erdős), gave bounds, and conjectured the formula which Unger later proved.

Also the edge chromatic number of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 23 2018

References

Martin Aigner and Gunter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin, 2000. Chapter 10.

Links

Vincenzo Librandi, Table of n, a(n) for n = 3..1000

J. E. Goodman and R. Pollack, On the combinatorial classification of nondegenerate configurations in the plane, J. Combin. Theory Ser. A, 29 (1980), pp. 220-235.

Giovanni Resta, Illustration of terms a(3)-a(7)

P. R. Scott, On the sets of directions determined by n points, The American Mathematical Monthly 77:5 (1970), pp. 502-505.

Peter Ungar, 2N noncollinear points determine at least 2N directions, Journal of Combinatorial Theory, Series A, 33:3 (1982), pp. 343-347.

Eric Weisstein's World of Mathematics, Edge Chromatic Number

Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

a(n) = 2*floor(n/2) for n > 3.

G.f.: x^3*(3+x-3*x^2+x^3)/((1+x)*(1-x)^2). [Bruno Berselli, Mar 04 2013]

Maple

A213222:=n->`if`(n = 3, 3, 2*floor(n/2)); seq(A213222(n), n=3..100); # Wesley Ivan Hurt, Mar 28 2014

Mathematica

CoefficientList[Series[(3 + x - 3 x^2 + x^3)/((1 + x) (1 - x)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 29 2014 *)

Prog

(PARI) a(n)=if(n>3, n\2*2, 3)
(Magma) [2*Floor(n/2): n in [3..100]]; // Vincenzo Librandi, Mar 29 2014

Crossrefs

Cf. A000769, A000755, A219760.

Cf. A000217 (maximum number of slopes, with offset 1).

Sequence in context: A354154 A229022 A014683 * A166737 A088847 A120613

Adjacent sequences: A213219 A213220 A213221 * A213223 A213224 A213225

Keyword

nonn,easy

Author

Charles R Greathouse IV, Mar 02 2013

Status

approved