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

1,1

Comments

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

Links

Table of n, a(n) for n=1..22.

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).

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

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]

Crossrefs

Sequence in context: A367507 A303602 A000544 * A221777 A227995 A037776

Adjacent sequences: A356197 A356198 A356199 * A356201 A356202 A356203

Keyword

nonn,easy

Author

Eric W. Weisstein, Jul 29 2022

Status

approved