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 Dominating Set
Index entries for linear recurrences with constant coefficients, signature (4, -3, -4, 4, -1, 1, 3, -3, 0, 2, 3, 0, -1).
FORMULA
From Andrew Howroyd, Aug 27 2017: (Start)
a(n) = 4*a(n-1) - 3*a(n-2) - 4*a(n-3) + 4*a(n-4) - a(n-5) + a(n-6) + 3*a(n-7) - 3*a(n-8) + 2*a(n-10) + 3*a(n-11) - a(n-13) for n > 13.
G.f.: x*(2 - 18*x^2 + 16*x^3 + 21*x^4 - 18*x^5 - 15*x^6 - 2*x^7 + 16*x^8 + 2*x^9 + 11*x^10 - 2*x^11 - 5*x^12)/((1 + x^2)*(1 - x - x^2)*(1 - x - x^2 - x^3)*(1 - 2*x - x^2 + 3*x^3 - x^4 - 2*x^5 + x^6)).
(End)
MATHEMATICA
Table[RootSum[-1 - # - #^2 + #^3 &, #^n &] + RootSum[1 - 2 # - #^2 + 3 #^3 - #^4 - 2 #^5 + #^6 &, #^n &] - LucasL[n] - 2 Cos[n Pi/2], {n, 20}]
LinearRecurrence[{4, -3, -4, 4, -1, 1, 3, -3, 0, 2, 3, 0, -1}, {2, 8, 8, 16, 37, 80, 156, 304, 602, 1173, 2290, 4456, 8686}, 20]
CoefficientList[Series[(2 - 18 x^2 + 16 x^3 + 21 x^4 - 18 x^5 - 15 x^6 - 2 x^7 + 16 x^8 + 2 x^9 + 11 x^10 - 2 x^11 - 5 x^12)/((1 + x^2) (1 - x - x^2) (1 - x - x^2 - x^3) (1 - 2 x - x^2 + 3 x^3 - x^4 - 2 x^5 + x^6)), {x, 0, 20}], x]
PROG
(PARI) Vec((2 - 18*x^2 + 16*x^3 + 21*x^4 - 18*x^5 - 15*x^6 - 2*x^7 + 16*x^8 + 2*x^9 + 11*x^10 - 2*x^11 - 5*x^12)/((1 + x^2)*(1 - x - x^2)*(1 - x - x^2 - x^3)*(1 - 2*x - x^2 + 3*x^3 - x^4 - 2*x^5 + x^6)) + O(x^30)) \\ Andrew Howroyd, Aug 27 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jul 28 2017
EXTENSIONS
a(13)-a(24) from Andrew Howroyd, Aug 11 2017
a(1)-a(2) and terms a(25) and beyond from Andrew Howroyd, Aug 27 2017
STATUS
approved