A297047 - OEIS

%I #24 Jun 28 2024 09:29:32

%S 0,2,10,41,154,562,2023,7240,25842,92129,328270,1169390,4165231,

%T 14835316,52837774,188186161,670237602,2387090906,8501757271,

%U 30279468752,107841945274,384084812929,1367938393414,4871984909782,17351831683935,61799465142812,220102059235510

%N Number of edge covers in the n-wheel graph.

%C Extended to a(1)-a(3) using the formula/recurrence.

%H Robert Israel, <a href="/A297047/b297047.txt">Table of n, a(n) for n = 1..1811</a>

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

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-5,-2).

%F a(n) = A206776(n-1) - A000032(n-1).

%F a(n) = 4*a(n-1) - 5*a(n-3) - 2*a(n-4).

%F G.f.: x^2*(2+2*x+x^2) / ( (x^2+x-1)*(2*x^2+3*x-1) ).

%F a(n) = 2^(-2-n)*(2*(1-sqrt(5))^n*(1+sqrt(5)) - 2*(-1+sqrt(5))*(1+sqrt(5))^n - 3*(3-sqrt(17))^n-sqrt(17)*(3-sqrt(17))^n - 3*(3+sqrt(17))^n+sqrt(17)*(3+sqrt(17))^n). - _Colin Barker_, Dec 28 2017

%p f:= gfun:-rectoproc({a(n) = 4*a(n-1) - 5*a(n-3) - 2*a(n-4),a(1)=0,a(2)=2,a(3)=10,a(4)=41},a(n),remember):

%p map(f, [$1..30]); # _Robert Israel_, Dec 26 2017

%t Table[I^(n - 1) 2^((n + 1)/2) ChebyshevT[n - 1, -3 I/(2 Sqrt[2])] - LucasL[n - 1, 1], {n, 20}]

%t LinearRecurrence[{4, 0, -5, -2}, {0, 2, 10, 41}, 20]

%t CoefficientList[Series[x (2 + 2 x + x^2)/(1 - 4 x + 5 x^3 + 2 x^4), {x, 0, 20}], x]

%o (PARI) first(n) = Vec(x^2*(2 + 2*x + x^2)/(1 - 4*x + 5*x^3 + 2*x^4) + O(x^(n+1)), -n) \\ _Iain Fox_, Dec 24 2017

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_, Dec 24 2017