OFFSET
1,1
COMMENTS
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).
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..800
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 (26,-196,486,-315).
FORMULA
a(n) = 8*15^n - 12*7^n + 9*3^n - 4.
From Stefano Spezia, Jul 08 2024: (Start)
G.f.: x*(59 - 245*x + 1173*x^2 - 315*x^3)/((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)).
E.g.f.: 8*exp(15*x) - 12*exp(7*x) + 9*exp(3*x) - 4*exp(x) - 1. (End)
MATHEMATICA
LinearRecurrence[{26, -196, 486, -315}, {59, 1289, 23123, 376913}, 20] (* Paolo Xausa, Jan 22 2025 *)
PROG
(Python)
def a_n(n):
return 8 * 15**n - 12*7^n + 9 * 3**n + 3
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Marshall Nicholson, Jul 08 2024
STATUS
approved