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Number of irredundant sets in the n-ladder graph.
1

%I #15 Aug 06 2017 22:46:58

%S 3,11,26,79,224,640,1828,5225,14928,42654,121873,348232,995003,

%T 2843014,8123337,23210809,66320216,189496620,541448364,1547079580,

%U 4420468031,12630596045,36089381477,103118131368,294639269914,841872308017,2405480380385,6873175192304

%N Number of irredundant sets in the n-ladder graph.

%C Row 2 of A286868.

%H Andrew Howroyd, <a href="/A290513/b290513.txt">Table of n, a(n) for n = 1..200</a>

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

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

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, 3, 5, -2, -6, -5, -4, -4, -3, -2, -2).

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

%F a(n) = 2*a(n-1) + a(n-2) + 3*a(n-3) + 5*a(n-4) - 2*a(n-5) - 6*a(n-6) - 5*a(n-7) - 4*a(n-8) - 4*a(n-9) - 3*a(n-10) - 2*a(n-11) - 2*a(n-12).

%F G.f.: x*(3 + 5*x + x^2 + 7*x^3 - 8*x^4 - 14*x^5 - 3*x^6 - 5*x^7 - 9*x^8 - 3*x^9 - 2*x^10 - 2*x^11) / (1 - 2*x - x^2 - 3*x^3 - 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 4*x^8 + 4*x^9 + 3*x^10 + 2*x^11 + 2*x^12).

%F (End)

%t Table[RootSum[2 + 2 # + 3 #^2 + 4 #^3 + 4 #^4 + 5 #^5 + 6 #^6 + 2 #^7 - 5 #^8 - 3 #^9 - #^10 - 2 #^11 + #^12 &, -105577159431355949 #^n - 389671420034091247 #^(n + 1) + 29241021604101932 #^(n + 2) + 551171239538727862 #^(n + 3) - 835076245333578054 #^(n + 4) - 823168743731791895 #^(n + 5) + 1288188291747539683 #^(n + 6) + 114497498217658607 #^(n + 7) - 529545500369064866 #^(n + 8) + 763912087212707104 #^(n + 9) - 593709228550713556 #^(n + 10) + 130400048784652699 #^(n + 11) &]/2485579507903393779, {n, 20}] (* _Eric W. Weisstein_, Aug 05 2017 *)

%t LinearRecurrence[{2, 1, 3, 5, -2, -6, -5, -4, -4, -3, -2, -2}, {3, 11, 26, 79, 224, 640, 1828, 5225, 14928, 42654, 121873, 348232}, 20] (* _Eric W. Weisstein_, Aug 05 2017 *)

%t CoefficientList[Series[(3 + 5 x + x^2 + 7 x^3 - 8 x^4 - 14 x^5 - 3 x^6 - 5 x^7 - 9 x^8 - 3 x^9 - 2 x^10 - 2 x^11)/(1 - 2 x - x^2 - 3 x^3 - 5 x^4 + 2 x^5 + 6 x^6 + 5 x^7 + 4 x^8 + 4 x^9 + 3 x^10 + 2 x^11 + 2 x^12), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 05 2017 *)

%o (PARI) Vec((3+5*x+x^2+7*x^3-8*x^4-14*x^5-3*x^6-5*x^7-9*x^8-3*x^9-2*x^10-2*x^11)/(1-2*x-x^2-3*x^3-5*x^4+2*x^5+6*x^6+5*x^7+4*x^8+4*x^9+3*x^10+2*x^11+2*x^12)+O(x^30)) \\ _Andrew Howroyd_, Aug 04 2017

%Y Cf. A286868 (irredundant sets in m X n grid graph).

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Aug 04 2017

%E Terms a(11) and beyond from _Andrew Howroyd_, Aug 04 2017