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 A291055 #22 Feb 16 2025 08:33:50
%S A291055 1,2,2,4,6,8,13,17,27,40,57,86,122,184,269,395,582,849,1255,1843,2708,
%T A291055 3982,5841,8597,12631,18566,27286,40082,58929,86598,127279,187052,
%U A291055 274872,404001,593732,872606,1282416,1884660,2769856,4070718,5982611,8792345
%N A291055 Number of maximal irredundant sets in the n-path graph.
%C A291055 From _Andrew Howroyd_, Aug 23 2017: (Start)
%C A291055 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 second if n is a multiple of 3, otherwise starting with the first.
%C A291055 The maximum size of an irredundant set, the upper irredundance number, is given by ceiling(n/2). A suitable construction for such a set is every second vertex starting with the first.
%C A291055 (End)
%H A291055 Andrew Howroyd, <a href="/A291055/b291055.txt">Table of n, a(n) for n = 1..500</a>
%H A291055 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H A291055 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>
%H A291055 <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 A291055 From _Andrew Howroyd_, Aug 23 2017: (Start)
%F A291055 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 A291055 G.f.: x*(1 + 2*x + x^2 + x^3 + x^4 - x^5 - x^6 - 2*x^7 + 3*x^9 - x^12 - x^13)/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + x^14).
%F A291055 (End)
%e A291055 Case n=5: maximal irredundant sets represented as binary words are {00110, 01001, 01010, 01100, 10010, 10101}, so a(5)=6. - _Andrew Howroyd_, Aug 23 2017
%t A291055 Rest @ CoefficientList[Series[x (1 + 2 x + x^2 + x^3 + x^4 - x^5 - x^6 - 2 x^7 + 3 x^9 - x^12 - x^13)/(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, 42}], x] (* _Michael De Vlieger_, Aug 24 2017 *)
%t A291055 LinearRecurrence[{0, 1, 1, 1, 1, 0, -1, -2, -1, 2, 1, 0, 0, -1}, {1, 2, 2, 4, 6, 8, 13, 17, 27, 40, 57, 86, 122, 184}, 20] (* _Eric W. Weisstein_, Aug 28 2017 *)
%t A291055 RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, -4480566127993567 #^n + 2115784835595702 #^(n+1) - 8803686900182082 #^(n+2) + 12438105918248674 #^(n+3) + 9975829435558087 #^(n+4) + 32647411155695559 #^(n+5) + 921201586573742 #^(n+6) - 12400355965941932 #^(n+7) - 18709447182799197 #^(n+8) - 16194871035876814 #^(n+9) - 8478829128434826 #^(n+10) - 3824486277258301 #^(n+11) + 902031297001609 #^(n+12) + 11119370357865554 #^(n+13) &]/333325507942333403 (* _Eric W. Weisstein_, Aug 28 2017 *)
%o A291055 (PARI) Vec((1 + 2*x + x^2 + x^3 + x^4 - x^5 - x^6 - 2*x^7 + 3*x^9 - x^12 - x^13)/(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 23 2017
%Y A291055 Row 1 of A291439.
%Y A291055 Row sums of A291375.
%Y A291055 Cf. A286887, A286954, A291444.
%K A291055 nonn
%O A291055 1,2
%A A291055 _Eric W. Weisstein_, Aug 17 2017
%E A291055 Terms a(21) and beyond from _Andrew Howroyd_, Aug 23 2017