OFFSET
1,2
COMMENTS
Number of connected dominating sets in the n-cocktail party graph.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Connected Dominating Set
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).
FORMULA
a(n) = 4^n - 3*n - 1.
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-2).
G.f.: 9*x^2/((1-x)^2*(1-4*x)).
E.g.f.: exp(4*x) - (1+3*x)*exp(x). - G. C. Greubel, Mar 26 2022
MATHEMATICA
Table[4^n - 3 n - 1, {n, 20}]
LinearRecurrence[{6, -9, 4}, {0, 9, 54}, 20]
CoefficientList[Series[-((9 x)/((-1 + x)^2 (-1 + 4 x))), {x, 0, 20}], x]
PROG
(PARI) concat(0, Vec(9*x^2 / ((1 - x)^2*(1 - 4*x)) + O(x^30))) \\ Colin Barker, Jun 30 2017
(Magma) [4^n -(3*n+1): n in [1..30]]; // G. C. Greubel, Mar 26 2022
(Sage) [4^n -(3*n+1) for n in (1..30)] # G. C. Greubel, Mar 26 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Jun 29 2017
STATUS
approved