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A291048 Number of nonequivalent maximal irredundant sets in the n-cycle graph up to rotation.

%I A291048 #19 May 29 2025 10:40:10
%S A291048 0,1,1,2,2,3,2,3,4,8,6,11,11,17,25,32,41,59,79,118,157,221,303,436,
%T A291048 610,864,1215,1724,2436,3484,4926,7029,9990,14270,20354,29113,41572,
%U A291048 59517,85186,122127,175018,251176,360404,517758,743895,1069633,1538313,2213894
%N A291048 Number of nonequivalent maximal irredundant sets in the n-cycle graph up to rotation.
%C A291048 Equivalently, the number of n-bead binary necklaces (with turnover not allowed) avoiding the patterns 111, 1101, 1011, 00000, 000010, 010000, 000100, 001000, 0100010.
%H A291048 Andrew Howroyd, <a href="/A291048/b291048.txt">Table of n, a(n) for n = 1..200</a>
%H A291048 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>.
%H A291048 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>.
%F A291048 a(n) = (1/n) * Sum_{d|n} phi(n/d) * A286954(d).
%e A291048 Case n=7: admissible nonequivalent words are 0010011 and 0010101, so a(7)=2.
%t A291048 Table[(1/n) Sum[EulerPhi[n/d] SeriesCoefficient[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), {x, 0, d}], {d, Divisors@ n}], {n, 48}] (* _Michael De Vlieger_, Aug 17 2017 *)
%o A291048 (PARI)
%o A291048 {my (v=concat([0],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^50))));
%o A291048 vector(length(v), n, sumdiv(n,d,eulerphi(n/d)*v[d])/n)}
%Y A291048 Cf. A286954.
%K A291048 nonn
%O A291048 1,4
%A A291048 _Andrew Howroyd_, Aug 16 2017