A320604 Chromatic number of the n-polygon diagonal intersection graph.

4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3

Offset

1,1

Comments

Extended to a(1)-a(2) using the formula/recurrence.

Links

Table of n, a(n) for n=1..92.

Eric Weisstein's World of Mathematics, Chromatic Number

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

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1).

Formula

a(n) = (10 - (-1)^n + cos(n*Pi/3) - cos(2*n*Pi/3))/3.

a(n) = a(n-6).

G.f.: -x^3*(3 + 3*x + 4*x^2 + 3*x^3 + 4*x^4 + 3*x^5)/((-1 + x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).

E.g.f.: (1/6)*exp(-x)*(-2-18*exp(x)+20*exp(2*x)+exp((1+(-1)^(1/3))*x)-exp((1+(-1)^(2/3))*x)-exp(x-(-1)^(1/3)*x)+exp(x-(-1)^(2/3)*x)). - Stefano Spezia, Oct 17 2018

Mathematica

Table[(10 - (-1)^n + Cos[n Pi/3] - Cos[2 n Pi/3])/3, {n, 20}]
Table[Piecewise[{{4, Mod[n, 6] == 1 || Mod[n, 6] == 5}}, 3], {n, 20}]
LinearRecurrence[{0, 0, 0, 0, 0, 1}, {4, 3, 3, 3, 4, 3}, 20]
CoefficientList[Series[-((3 + 3 x + 4 x^2 + 3 x^3 + 4 x^4 + 3 x^5)/((-1 + x) (1 + x) (1 - x + x^2) (1 + x + x^2))), {x, 0, 20}], x]
Simplify[CoefficientList[Series[1/6 E^-x (-2 - 18 E^x + 20 E^(2 x) + E^((1 + (-1)^(1/3)) x) - E^((1 + (-1)^(2/3)) x) - E^(x - (-1)^(1/3) x) + E^(x - (-1)^(2/3) x))/x, {x, 0, 50}], x]*Table[(k+1)!, {k, 0, 50}]] (* Stefano Spezia, Oct 17 2018 *)

Crossrefs

Sequence in context: A085415 A187470 A059124 * A282445 A281705 A026858

Adjacent sequences: A320601 A320602 A320603 * A320605 A320606 A320607

Keyword

nonn

Author

Eric W. Weisstein, Oct 17 2018

Status

approved