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 A286954 Number of maximal irredundant sets in the n-cycle graph. 6
 0, 2, 3, 6, 10, 11, 14, 14, 30, 62, 66, 99, 143, 212, 343, 478, 697, 992, 1501, 2246, 3251, 4776, 6969, 10283, 15210, 22297, 32727, 47984, 70644, 103961, 152706, 224382, 329508, 484451, 712274, 1046736, 1538164, 2260109, 3321932, 4882574, 7175738, 10545581 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Andrew Howroyd, Aug 16 2017: (Start) Equivalently, the number of binary words not cyclically containing a sub-word of 111, 1101, 1011, 00000, 000010, 010000, 000100, 001000, 0100010. 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. 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. (End) LINKS Andrew Howroyd, Table of n, a(n) for n = 1..500 Eric Weisstein's World of Mathematics, Cycle Graph. Eric Weisstein's World of Mathematics, Maximal Irredundant Set. Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,1,0,-1,-2,-1,2,1,0,0,-1). FORMULA From Andrew Howroyd, Aug 16 2017: (Start) 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. 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). a(p) = p * A291048(p) for p prime. (End) EXAMPLE From Andrew Howroyd, Aug 16 2017: (Start) Case n=2: admissible words are 01 and 10, so a(2)=2. Case n=3: admissible words are 001, 010, 100, so a(3)=3. Case n=7: up to rotation, admissible words are 0010011 and 0010101, so a(7) = 7*2 = 14. (End) MATHEMATICA 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] 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] Table[RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, #^n &], {n, 20}] RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, #^Range[20] &] PROG (PARI) 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 CROSSREFS Row sums of A291044. Cf. A290493, A291048. Sequence in context: A342766 A048633 A225175 * A047402 A088196 A112925 Adjacent sequences: A286951 A286952 A286953 * A286955 A286956 A286957 KEYWORD nonn AUTHOR Eric W. Weisstein, Aug 15 2017 EXTENSIONS a(1)-a(2) and terms a(21) and beyond from Andrew Howroyd, Aug 16 2017 STATUS approved

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Last modified September 9 21:06 EDT 2024. Contains 375765 sequences. (Running on oeis4.)