A290762 Number of minimal edge covers in the n-gear graph.

2, 6, 15, 38, 90, 200, 434, 934, 1995, 4237, 8976, 19010, 40300, 85591, 182231, 389094, 833306, 1790141, 3857190, 8334719, 18057605, 39217422, 85357692, 186142694, 406619110, 889555565, 1948564239, 4273011841, 9379101468, 20603197661, 45289832230, 99612356518

Offset

1,1

Comments

Sequence extrapolated to n = 1 using recurrence. - Andrew Howroyd, Aug 27 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Gear Graph

Eric Weisstein's World of Mathematics, Minimal Edge Cover

Index entries for linear recurrences with constant coefficients, signature (6, -14, 19, -21, 18, -11, 7, -2, 1).

Formula

From Andrew Howroyd, Aug 27 2017: (Start)

a(n) = 6*a(n-1) - 14*a(n-2) + 19*a(n-3) - 21*a(n-4) + 18*a(n-5) - 11*a(n-6) + 7*a(n-7) - 2*a(n-8) + a(n-9) for n > 9.

G.f.: x*(2 - 6*x + 7*x^2 - 6*x^3 - 3*x^5 + x^6)/((1 - 2*x + x^2 - x^3)^2*(1 - 2*x - x^3)).

(End)

Mathematica

CoefficientList[Series[(-2 + 6 x - 7 x^2 + 6 x^3 + 3 x^5 - x^6)/((-1 + 2 x + x^3) (-1 + 2 x - x^2 + x^3)^2), {x, 0, 20}], x]
LinearRecurrence[{6, -14, 19, -21, 18, -11, 7, -2, 1}, {2, 6, 15, 38, 90, 200, 434, 934, 1995}, 20]
Table[RootSum[-1 - 2 #^2 + #^3 &, -2 #^(n+2) + #^(n+3) &] - RootSum[-1 + # - 2 #^2 + #^3 &, #^(n+1) - 2 #^(n+2) + #^(n+3) &] + n RootSum[-1 + # - 2 #^2 + #^3 &, -7 #^(n+1) - 14 #^(n+2) + 13 #^(n+3) &]/23, {n, 20}]

Prog

(PARI) Vec((2 - 6*x + 7*x^2 - 6*x^3 - 3*x^5 + x^6)/((1 - 2*x + x^2 - x^3)^2*(1 - 2*x - x^3)) + O(x^40)) \\ Andrew Howroyd, Aug 27 2017

Crossrefs

Cf. A287349, A287425.

Sequence in context: A306463 A034518 A260787 * A106515 A153122 A109545

Adjacent sequences: A290759 A290760 A290761 * A290763 A290764 A290765

Keyword

nonn

Author

Eric W. Weisstein, Aug 10 2017

Extensions

a(9)-a(20) from Andrew Howroyd, Aug 10 2017

a(1)-a(2) and terms a(21) and beyond from Andrew Howroyd, Aug 27 2017

Status

approved