OFFSET
1,1
COMMENTS
a(n) is the number of connected dominating sets and total dominating sets in the n-path complement graph for n > 1.
LINKS
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Path Complement Graph.
Eric Weisstein's World of Mathematics, Total Dominating Set.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
FORMULA
G.f.: -2*x*(1 - 4*x + 5*x^2)/((-1 + x)^2*(-1 + 2*x)).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
a(n) = A258547(n-5) for n > 5 (conjectured).
a(n) = 4*A000295(n) for n > 1. - Alois P. Heinz, Apr 12 2018
From Elmo R. Oliveira, Nov 13 2025: (Start)
E.g.f.: exp(2*x) + 4*(1 - x)*exp(x) - 5.
a(n) = 2*A005803(n-1). (End)
MATHEMATICA
Table[4 + 2^n - 4 n, {n, 20}]
LinearRecurrence[{4, -5, 2}, {2, 0, 0}, 20]
CoefficientList[Series[-(2 (1 - 4 x + 5 x^2)/((-1 + x)^2 (-1 + 2 x))), {x, 0, 20}], x]
PROG
(PARI) a(n) = 4+2^n-4*n; \\ Altug Alkan, Apr 12 2018
(Magma) [4+2^n-4*n : n in [1..45]]; // Vincenzo Librandi, Apr 13 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Apr 11 2018
STATUS
approved