This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A286954 #47 Feb 16 2025 08:33:45
%S A286954 0,2,3,6,10,11,14,14,30,62,66,99,143,212,343,478,697,992,1501,2246,
%T A286954 3251,4776,6969,10283,15210,22297,32727,47984,70644,103961,152706,
%U A286954 224382,329508,484451,712274,1046736,1538164,2260109,3321932,4882574,7175738,10545581
%N A286954 Number of maximal irredundant sets in the n-cycle graph.
%C A286954 From _Andrew Howroyd_, Aug 16 2017: (Start)
%C A286954 Equivalently, the number of binary words not cyclically containing a sub-word of 111, 1101, 1011, 00000, 000010, 010000, 000100, 001000, 0100010.
%C A286954 For n > 1, the minimum size of a maximal irredundant set, the irredundance number, is given by ceiling(n/3). A suitable construction for such a set is every third vertex starting with the first.
%C A286954 The maximum size of an irredundant set, the upper irredundance number, is given by floor(n/2). For n > 1, a suitable construction for such a set is every second vertex starting with the second.
%C A286954 (End)
%H A286954 Andrew Howroyd, <a href="/A286954/b286954.txt">Table of n, a(n) for n = 1..500</a>
%H A286954 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>.
%H A286954 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>.
%H A286954 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,1,0,-1,-2,-1,2,1,0,0,-1).
%F A286954 From _Andrew Howroyd_, Aug 16 2017: (Start)
%F A286954 a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) - a(n-7) - 2*a(n-8) - a(n-9) + 2*a(n-10) + a(n-11) - a(n-14) for n > 14.
%F A286954 G.f.: x^2*(2 + 3*x + 4*x^2 + 5*x^3 - 7*x^5 - 16*x^6 - 9*x^7 + 20*x^8 + 11*x^9 - 14*x^12)/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + x^14).
%F A286954 a(p) = p * A291048(p) for p prime.
%F A286954 (End)
%e A286954 From _Andrew Howroyd_, Aug 16 2017: (Start)
%e A286954 Case n=2: admissible words are 01 and 10, so a(2)=2.
%e A286954 Case n=3: admissible words are 001, 010, 100, so a(3)=3.
%e A286954 Case n=7: up to rotation, admissible words are 0010011 and 0010101, so a(7) = 7*2 = 14.
%e A286954 (End)
%t A286954 LinearRecurrence[{0, 1, 1, 1, 1, 0, -1, -2, -1, 2, 1, 0, 0, -1}, {0, 2, 3, 6, 10, 11, 14, 14, 30, 62, 66, 99, 143, 212}, 20]
%t A286954 CoefficientList[Series[(x (2 + 3 x + 4 x^2 + 5 x^3 - 7 x^5 - 16 x^6 - 9 x^7 + 20 x^8 + 11 x^9 - 14 x^12))/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2 x^8 + x^9 - 2 x^10 - x^11 + x^14), {x, 0, 20}], x]
%t A286954 Table[RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, #^n &], {n, 20}]
%t A286954 RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, #^Range[20] &]
%o A286954 (PARI)
%o A286954 Vec((2 + 3*x + 4*x^2 + 5*x^3 - 7*x^5 - 16*x^6 - 9*x^7 + 20*x^8 + 11*x^9 - 14*x^12)/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + x^14)+O(x^40)) \\ _Andrew Howroyd_, Aug 16 2017
%Y A286954 Row sums of A291044.
%Y A286954 Cf. A290493, A291048.
%K A286954 nonn
%O A286954 1,2
%A A286954 _Eric W. Weisstein_, Aug 15 2017
%E A286954 a(1)-a(2) and terms a(21) and beyond from _Andrew Howroyd_, Aug 16 2017