A297047 Number of edge covers in the n-wheel graph.

0, 2, 10, 41, 154, 562, 2023, 7240, 25842, 92129, 328270, 1169390, 4165231, 14835316, 52837774, 188186161, 670237602, 2387090906, 8501757271, 30279468752, 107841945274, 384084812929, 1367938393414, 4871984909782, 17351831683935, 61799465142812, 220102059235510

Offset: 1

Author

Eric W. Weisstein, Dec 24 2017

Keywords

• nonn
• easy

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 1..1811
• Eric Weisstein's World of Mathematics, Edge Cover
• Eric Weisstein's World of Mathematics, Wheel Graph
• Index entries for linear recurrences with constant coefficients, signature (4,0,-5,-2).

Programs

Maple

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):
map(f, [$1..30]); # Robert Israel, Dec 26 2017

Mathematica

Table[I^(n - 1) 2^((n + 1)/2) ChebyshevT[n - 1, -3 I/(2 Sqrt[2])] - LucasL[n - 1, 1], {n, 20}]
LinearRecurrence[{4, 0, -5, -2}, {0, 2, 10, 41}, 20]
CoefficientList[Series[x (2 + 2 x + x^2)/(1 - 4 x + 5 x^3 + 2 x^4), {x, 0, 20}], x]

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

Formula

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

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

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

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