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Number of minimal dominating sets in the n-gear graph.
4

%I #25 Mar 04 2024 00:03:19

%S 2,8,8,16,37,80,156,304,602,1173,2290,4456,8686,16892,32833,63776,

%T 123864,240524,467060,907061,1761894,3423164,6652706,12933280,

%U 25151787,48931280,95228360,185400336,361093444,703546005,1371282460,2673742784,5215147858

%N Number of minimal dominating sets in the n-gear graph.

%C Sequence extrapolated to n = 1 using recurrence. - _Andrew Howroyd_, Aug 27 2017

%H Andrew Howroyd, <a href="/A290378/b290378.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalDominatingSet.html">Minimal Dominating Set</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (4, -3, -4, 4, -1, 1, 3, -3, 0, 2, 3, 0, -1).

%F From _Andrew Howroyd_, Aug 27 2017: (Start)

%F 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.

%F 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)).

%F (End)

%t 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}]

%t 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]

%t 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]

%o (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

%Y Cf. A290589, A290938.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Jul 28 2017

%E a(13)-a(24) from _Andrew Howroyd_, Aug 11 2017

%E a(1)-a(2) and terms a(25) and beyond from _Andrew Howroyd_, Aug 27 2017