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%I #22 Aug 04 2017 15:19:39
%S 2,3,5,9,15,26,44,76,130,223,382,655,1123,1925,3300,5657,9698,16625,
%T 28500,48857,83755,143580,246137,421949,723341,1240013,2125736,
%U 3644118,6247058,10709240,18358693,31472038,53952053,92489213,158552901,271804912,465951173
%N Number of irredundant sets in the path graph P_n.
%C Equivalently, the number of binary words of length n that don't start or end with 11 (the outside 1 is redundant) and don't contain 111, 1101 or 1011 (the middle 1 is redundant).
%H Andrew Howroyd, <a href="/A286887/b286887.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/PathGraph.html">Path Graph</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,1,0,-1).
%F a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-6) for n > 6.
%F G.f.: x*(1 + x)*(2 - x + x^2 - x^4)/(1 - x - x^2 - x^4 + x^6).
%e Case n=5: irredundant words are {00000, 00001, 00010, 00100, 01000, 10000, 00110, 01100, 10001, 00101, 01010, 10100, 01001, 10010, 10101}, so a(5)=15.
%t RootSum[1 - #^2 - #^4 - #^5 + #^6 &, 3191 #^n + 4752 #^(1 + n) - 4234 #^(2 + n) + 11985 #^(3 + n) - 2369 #^(4 + n) + 3536 #^(5 + n) &]/89653 (* _Eric W. Weisstein_, Aug 04 2017 *)
%t LinearRecurrence[{1, 1, 0, 1, 0, -1}, {2, 3, 5, 9, 15, 26}, 20] (* _Eric W. Weisstein_, Aug 04 2017 *)
%t CoefficientList[Series[(2 + x + x^3 - x^4 - x^5)/(1 - x - x^2 - x^4 + x^6), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 04 2017 *)
%o (PARI) Vec((1 + x)*(2 - x + x^2 - x^4)/(1 - x - x^2 - x^4 + x^6) + O(x^40))
%Y Row 1 of A286868 and A286870.
%K nonn,easy
%O 1,1
%A _Andrew Howroyd_, Aug 02 2017
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