A373293 - OEIS

%I #42 Jul 24 2024 17:33:00

%S 16,154,1240,9202,66016,466954,3283240,23026402,161316016,1129605754,

%T 7908421240,55362491602,387548070016,2712868376554,18990174295240,

%U 132931507044802,930521410248016,6513652454539354,45595574930185240,319169047756526002

%N Number of edge covers of the fan graph F_{n,3}.

%C Label vertices of F_{n,3} with v1, ..., v_n, a, b, c, with b adjacent to both a and c. An edge cover is a subset of the edges so that each vertex is the endpoint of at least one edge. Each of the n vertices has 7 ways to connect to vertices a, b, c. Once we connect all v_i's, we then have 4 options for the edges ab and bc to exist or not, giving 4*7^m options. But not all of these will be an edge cover. For example, if all v_i's connected to a and b only, we have to add edge bc in the second step. So 5*3^m are removed. But we removed 3 cases where all v_i's connected to only a, or only b, or only c too many times.

%H Paolo Xausa, <a href="/A373293/b373293.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-31,21).

%F a(n) = 4*7^n - 5*3^n + 3.

%F From _Stefano Spezia_, Jun 24 2024: (Start)

%F G.f.: 2*x*(8 - 11*x + 21*x^2)/((1 - x)*(1 - 3*x)*(1 - 7*x)).

%F E.g.f.: 4*exp(7*x) - 5*exp(3*x) + 3*exp(x) - 2. (End)

%t A373293[n_] := 4*7^n - 5*3^n + 3; Array[A373293, 25] (* or *)

%t LinearRecurrence[{11, -31, 21}, {16, 154, 1240}, 25] (* _Paolo Xausa_, Jun 24 2024 *)

%o (Python)

%o def a_n(n):

%o     return 4 * 7**n - 5 * 3**n + 3

%Y Cf. A100774 (in F_{n,2}).

%K nonn,easy

%O 1,1

%A _Feryal Alayont_, Jun 22 2024