OFFSET
1,1
COMMENTS
Extended to a(1)-a(2) using the formula/recurrence.
LINKS
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
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Oct 17 2018
STATUS
approved