A287063 - OEIS

%I #20 Sep 04 2021 20:08:27

%S 3,9,39,183,833,3629,15291,63051,256605,1036401,4167815,16720031,

%T 66986169,268173525,1073185011,4293787923,17177379125,68714234201,

%U 274866897279,1099488559527,4397998277073,17592085381629,70368534463019,281474540503643,1125899000873613

%N Number of dominating sets in the n-crown graph (for n > 1).

%C The 1-crown graph is the 2-empty graph bar K_2 which has a single dominating set and so differs from a(1) = 3. - _Eric W. Weisstein_, Sep 04 2021

%H Colin Barker, <a href="/A287063/b287063.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Crown Graph.html">Crown Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (11,-47,101,-116,68,-16).

%F a(n) = 4^n - 2^n*(n + 2) + n^2 + n + 3.

%F From _Colin Barker_, May 19 2017: (Start)

%F G.f.: x*(3 - 24*x + 81*x^2 - 126*x^3 + 92*x^4 - 32*x^5) / ((1 - x)^3*(1 - 2*x)^2*(1 - 4*x)).

%F a(n) = 11*a(n-1) - 47*a(n-2) + 101*a(n-3) - 116*a(n-4) + 68*a(n-5) - 16*a(n-6) for n>6.

%F (End)

%t Table[4^n - 2^n (n + 2) + n^2 + n + 3, {n, 25}]

%t LinearRecurrence[{11, -47, 101, -116, 68, -16}, {3, 9, 39, 183, 833, 3629}, 25]

%o (PARI) Vec( x*(3 - 24*x + 81*x^2 - 126*x^3 + 92*x^4 - 32*x^5) / ((1 - x)^3*(1 - 2*x)^2*(1 - 4*x)) + O(x^30)) \\ _Colin Barker_, May 19 2017

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, May 19 2017