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
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