A356200 Number of edge covers in the n-gear graph.

3, 25, 162, 969, 5613, 32062, 181989, 1030017, 5821902, 32886505, 185714829, 1048619646, 5920559661, 33426829321, 188721717102, 1065481514817, 6015458406741, 33961820796094, 191740095366885, 1082517159435249, 6111623364952302, 34504707439240921

Offset: 1

Author

Eric W. Weisstein, Jul 29 2022

Keywords

• nonn
• easy

Comments

Sequence extended to a(1) using the formula/recurrence.

Links

• Eric Weisstein's World of Mathematics, Edge Cover
• Eric Weisstein's World of Mathematics, Gear Graph
• Index entries for linear recurrences with constant coefficients, signature (9,-21,12,-2).

Programs

Mathematica

Table[(3 - Sqrt[7])^n + (3 + Sqrt[7])^n - LucasL[2 n], {n, 30}] // Expand
CoefficientList[Series[(3 - 2 x)/((1 - 3 x + x^2) (1 - 6 x + 2 x^2)), {x, 0, 20}], x]
LinearRecurrence[{9, -21, 12, -2}, {3, 25, 162, 969}, 20]

Formula

a(n) = (3-sqrt(7))^n + (sqrt(7)+3)^n - Lucas(2*n).

a(n) = 9*a(n-1) - 21*a(n-2) + 12*a(n-3) - 2*a(n-4).

G.f.: x*(3-2*x)/((1-3*x+x^2)*(1-6*x+2*x^2)).

a(n) = 2*A146963(n) - A005248(n). - R. J. Mathar, Jan 25 2023