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A290762 Number of minimal edge covers in the n-gear graph. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)