A287594 - OEIS

%I #13 Mar 06 2022 10:34:54

%S 3,4,12,28,72,184,480,1264,3360,8992,24192,65344,177024,480640,

%T 1307136,3559168,9699840,26452480,72173568,196989952,537802752,

%U 1468536832,4010582016,10954043392,29920862208,81733033984,223274237952,609947435008,1666309128192

%N Number of independent vertex sets in the n-helm graph.

%C Extended to a(0)-a(2) using the formula.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HelmGraph.html">Helm Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2, -4).

%F a(n) = 2^n+A080040(n).

%F a(n) = 2^n+(1-sqrt(3))^n+(1+sqrt(3))^n.

%F a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3).

%F G.f.: (3-8*x+2*x^2)/((1-2*x)*(1-2*x-2*x^2)).

%t Table[2^n + (1 - Sqrt[3])^n + (1 + Sqrt[3])^n, {n, 0, 20}] // Expand

%t Table[2^n + 2^(n/2) LucasL[n, Sqrt[2]], {n, 0, 20}] // Round

%t LinearRecurrence[{4, -2, -4}, {4, 12, 28}, {0, 20}]

%t CoefficientList[Series[(3 - 8 x + 2 x^2)/(1 - 4 x + 2 x^2 + 4 x^3), {x, 0, 20}], x]

%Y Cf. A080040.

%K nonn,easy

%O 0,1

%A _Eric W. Weisstein_, May 27 2017