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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
OFFSET
1,1
COMMENTS
Sequence extrapolated to n = 1 using recurrence. - Andrew Howroyd, Aug 27 2017
LINKS
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
Sequence in context: A306463 A034518 A260787 * A106515 A153122 A109545
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