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Number of minimal edge covers in the n-antiprism graph.
1

%I #14 Aug 11 2017 10:58:51

%S 2,14,74,286,1157,4778,19623,80478,330293,1355629,5563527,22832914,

%T 93707772,384582275,1578347684,6477630782,26584574434,109104640685,

%U 447771795953,1837681518261,7541951930181,30952609765223,127031312347552,521343900861138

%N Number of minimal edge covers in the n-antiprism graph.

%C The n-antiprism graph is well defined for n >= 3. Sequence extended to n = 1 using recurrence. - _Andrew Howroyd_, Aug 10 2017

%H Andrew Howroyd, <a href="/A290698/b290698.txt">Table of n, a(n) for n = 1..200</a>

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

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

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2, 5, 12, 11, 5, -3, -4, -1, 1).

%F From _Andrew Howroyd_, Aug 10 2017: (Start)

%F a(n) = 2*a(n-1) + 5*a(n-2) + 12*a(n-3) + 11*a(n-4) + 5*a(n-5) - 3*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).

%F G.f.: x*(2 + 10*x + 36*x^2 + 44*x^3 + 25*x^4 - 18*x^5 - 28*x^6 - 8*x^7 + 9*x^8)/((1 - 3*x - 3*x^2 - 6*x^3 - 2*x^4 + 3*x^5 + 2*x^6 - x^7)*(1 + x + x^2)).

%F (End)

%t Table[2 Cos[2 n Pi/3] + RootSum[-1 + 2 # + 3 #^2 - 2 #^3 - 6 #^4 - 3 #^5 - 3 #^6 + #^7 &, #^n &], {n, 20}]

%t LinearRecurrence[{2, 5, 12, 11, 5, -3, -4, -1, 1}, {2, 14, 74, 286, 1157, 4778, 19623, 80478, 330293}, 20]

%t CoefficientList[Series[(-2 - 10 x - 36 x^2 - 44 x^3 - 25 x^4 + 18 x^5 + 28 x^6 + 8 x^7 - 9 x^8)/(-1 + 2 x + 5 x^2 + 12 x^3 + 11 x^4 + 5 x^5 - 3 x^6 - 4 x^7 - x^8 + x^9), {x, 0, 20}], x]

%o (PARI)

%o Vec((2 + 10*x + 36*x^2 + 44*x^3 + 25*x^4 - 18*x^5 - 28*x^6 - 8*x^7 + 9*x^8)/((1 - 3*x - 3*x^2 - 6*x^3 - 2*x^4 + 3*x^5 + 2*x^6 - x^7)*(1 + x + x^2)) + O(x^30)) \\ _Andrew Howroyd_, Aug 10 2017

%Y Cf. A284700.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Aug 09 2017

%E a(1)-a(2) and terms a(7) and beyond from _Andrew Howroyd_, Aug 10 2017