login
a(n) = Lucas(n) + 3.
0

%I #9 Oct 12 2023 14:09:30

%S 4,6,7,10,14,21,32,50,79,126,202,325,524,846,1367,2210,3574,5781,9352,

%T 15130,24479,39606,64082,103685,167764,271446,439207,710650,1149854,

%U 1860501,3010352,4870850,7881199,12752046,20633242,33385285,54018524,87403806,141422327

%N a(n) = Lucas(n) + 3.

%C For n >= 3, number of independent vertex sets in the n-dipyramid graph.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DipyramidalGraph.html">Dipyramidal 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 (2,0,-1).

%F a(n) = A000032(n) + 3.

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

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

%t LucasL[Range[20]] + 3

%t Table[LucasL[n] + 3, {n, 20}]

%t LinearRecurrence[{2, 0, -1}, {4, 6, 7}, 20]

%t CoefficientList[Series[(4 - 2 x - 5 x^2)/(1 - 2 x + x^3), {x, 0, 20}], x]

%Y Cf. A000032 (Lucas numbers).

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, Oct 11 2023