OFFSET
1,1
COMMENTS
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.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Edge Cover.
Eric Weisstein's World of Mathematics, Fan Graph.
Index entries for linear recurrences with constant coefficients, signature (11,-31,21).
FORMULA
a(n) = 4*7^n - 5*3^n + 3.
From Stefano Spezia, Jun 24 2024: (Start)
G.f.: 2*x*(8 - 11*x + 21*x^2)/((1 - x)*(1 - 3*x)*(1 - 7*x)).
E.g.f.: 4*exp(7*x) - 5*exp(3*x) + 3*exp(x) - 2. (End)
MATHEMATICA
LinearRecurrence[{11, -31, 21}, {16, 154, 1240}, 25] (* Paolo Xausa, Jun 24 2024 *)
PROG
(Python)
def a_n(n):
return 4 * 7**n - 5 * 3**n + 3
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Feryal Alayont, Jun 22 2024
STATUS
approved