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

%I #19 Aug 10 2017 07:01:27

%S 1,5,7,17,31,52,99,169,301,536,947,1698,3043,5482,9919,17993,32743,

%T 59713,109099,199634,365719,670605,1230547,2259286,4149841,7624904,

%U 14013499,25759764,47358741,87077529,160120891,294453673,541509775,995888884,1831584239

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

%C The n-wheel graph is well defined for n>=4. Sequence extended to n=2 using recurrence. This gives values for a single edge and for a wheel with two spokes connected by two rim edges. - _Andrew Howroyd_, Aug 04 2017

%H Andrew Howroyd, <a href="/A290471/b290471.txt">Table of n, a(n) for n = 2..200</a>

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

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

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 3, 1, -5, -5, 0, 4, 3, 1).

%F From _Andrew Howroyd_, Aug 04 2017: (Start)

%F a(n) = a(n-1)+3*a(n-2)+a(n-3)-5*a(n-4)-5*a(n-5)+4*a(n-7)+3*a(n-8)+a(n-9) for n>10.

%F G.f.: x^2*(1 + x)*(1 + 3*x - 4*x^2 - 2*x^3 - 5*x^4 - 2*x^5 - x^6)/((1 - x^2 - x^3)^2*(1 - x - x^2 - x^3)).

%F (End)

%t LinearRecurrence[{1, 5, 7, 17, 31, 52, 99, 169, 301}, {1, 3, 1, -5, -5, 0, 4, 3, 1}, 20]

%t CoefficientList[Series[(-1 - 4 x + x^2 + 6 x^3 + 7 x^4 + 7 x^5 + 3 x^6 + x^7)/((-1 + x^2 + x^3)^2 (-1 + x + x^2 + x^3)), {x, 0, 20}], x]

%t Table[RootSum[-1 - # - #^2 + #^3 &, #^n &] - RootSum[-1 - # + #^3 &, -#^(1 + n) + #^(3 + n) &] + n RootSum[-1 - # + #^3 &, -#^(n + 1) - 8 #^(n + 2) + 13 #^(n + 3) &]/23, {n, 20}]

%o (PARI) Vec((1 + x)*(1 + 3*x - 4*x^2 - 2*x^3 - 5*x^4 - 2*x^5 - x^6)/((1 - x^2 - x^3)^2*(1 - x - x^2 - x^3))+O(x^40)) \\ _Andrew Howroyd_, Aug 04 2017

%K nonn

%O 2,2

%A _Eric W. Weisstein_, Aug 03 2017

%E a(2)-a(3) and terms a(14) and beyond from _Andrew Howroyd_, Aug 04 2017