A374450 - OEIS

%I #13 Jul 25 2024 14:10:35

%S 59,1289,23123,376913,5875499,89719769,1357012163,20434006433,

%T 307062808859,4609813953449,69174320548403,1037804612461553,

%U 15568397893099019,233535269569297529,3503094152437895843,52546868050923710273,788206211120541289979,11823115499323514984009,177346888817516282750483

%N Number of edge covers of the fan graph F_{n,4}.

%C Label vertices of F_{n,4} with v1, ..., v_n, a, b, c, d with be adjacent to a and c and d adjacent to 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 15 ways to connect to vertices a, b, c, d. Once we connect all v_i's, we then have 8 options for the edges ab, bc, and cd to exist or not, giving 8*15^n options. But not all of these will be an edge cover. For example, if all v_i's connect to only b and c, we have to add edges ab and cd in the second step. So, 12*7^n options are removed. However, we removed too many options here. This line of reasoning continues, following the Inclusion-Exclusion Principle, to obtain the remaining pieces of the formula for a(n).

%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_04">Index entries for linear recurrences with constant coefficients</a>, signature (26,-196,486,-315).

%F a(n) = 8*15^n - 12*7^n + 9*3^n - 4.

%F From _Stefano Spezia_, Jul 08 2024: (Start)

%F G.f.: x*(59 - 245*x + 1173*x^2 - 315*x^3)/((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)).

%F E.g.f.: 8*exp(15*x) - 12*exp(7*x) + 9*exp(3*x) - 4*exp(x) - 1. (End)

%o (Python)

%o def a_n(n):

%o     return 8 * 15**n - 12*7^n + 9 * 3**n + 3

%Y Cf. A100774, A373293, A374096.

%K nonn,easy

%O 1,1

%A _Marshall Nicholson_, Jul 08 2024