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%I #8 Oct 18 2018 03:29:49
%S 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,
%T 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,
%U 3,3,4,3,4,3,3,3,4,3,4,3,3,3,4,3,4,3,3,3,4,3,4,3
%N Chromatic number of the n-polygon diagonal intersection graph.
%C Extended to a(1)-a(2) using the formula/recurrence.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticNumber.html">Chromatic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html">Polygon Diagonal Intersection Graph</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 1).
%F a(n) = (10 - (-1)^n + cos(n*Pi/3) - cos(2*n*Pi/3))/3.
%F a(n) = a(n-6).
%F 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)).
%F 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
%t Table[(10 - (-1)^n + Cos[n Pi/3] - Cos[2 n Pi/3])/3, {n, 20}]
%t Table[Piecewise[{{4, Mod[n, 6] == 1 || Mod[n, 6] == 5}}, 3], {n, 20}]
%t LinearRecurrence[{0, 0, 0, 0, 0, 1}, {4, 3, 3, 3, 4, 3}, 20]
%t 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]
%t 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 *)
%K nonn
%O 1,1
%A _Eric W. Weisstein_, Oct 17 2018
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