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A301775 Number of odd chordless cycles in the (2n+1)-web graph.

%I A301775 #25 Feb 16 2025 08:33:53
%S A301775 0,12,30,74,200,522,1362,3572,9350,24474,64080,167762,439202,1149852,
%T A301775 3010350,7881194,20633240,54018522,141422322,370248452,969323030,
%U A301775 2537720634,6643838880,17393796002,45537549122,119218851372,312119004990,817138163594,2139295485800
%N A301775 Number of odd chordless cycles in the (2n+1)-web graph.
%H A301775 Andrew Howroyd, <a href="/A301775/b301775.txt">Table of n, a(n) for n = 1..200</a>
%H A301775 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>
%H A301775 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WebGraph.html">Web Graph</a>
%H A301775 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,2,-1).
%F A301775 From _Andrew Howroyd_, Mar 26 2018: (Start)
%F A301775 a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4) for n > 5.
%F A301775 G.f.: 2*x^2*(6 + 3*x + x^2 - x^3)/((1 - 3*x + x^2)*(1 + x + x^2)).
%F A301775 (End)
%F A301775 a(n) = A002878(n) + cos(2*n*Pi/3) - sqrt(3)*sin(2*n*Pi/3) for n > 1. - _Eric W. Weisstein_, Mar 27 2018
%F A301775 a(n) = A002878(n) + A131713(n), n>1. - _R. J. Mathar_, Apr 17 2018
%t A301775 Rest @ CoefficientList[Series[2 x^2*(6 + 3 x + x^2 - x^3)/((1 - 3 x + x^2) (1 + x + x^2)), {x, 0, 29}], x] (* _Michael De Vlieger_, Mar 26 2018 *)
%t A301775 Join[{0}, LinearRecurrence[{2, 1, 2, -1}, {12, 30, 74, 200}, 30]] (* _Vincenzo Librandi_, Mar 27 2018 *)
%t A301775 Join[{0}, Table[LucasL[2 n + 1] + Cos[2 n Pi/3] - Sqrt[3] Sin[2 n Pi/3], {n, 2, 20}]] (* _Eric W. Weisstein_, Mar 27 2018 *)
%o A301775 (PARI) Vec(2*(6 + 3*x + x^2 - x^3)/((1 - 3*x + x^2)*(1 + x + x^2)) + O(x^30)) \\ _Andrew Howroyd_, Mar 26 2018
%o A301775 (Magma) I:=[0,12,30,74,200]; [n le 5 select I[n] else  2*Self(n-1)+Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Mar 27 2018
%Y A301775 Cf. A301774.
%K A301775 nonn,easy
%O A301775 1,2
%A A301775 _Eric W. Weisstein_, Mar 26 2018
%E A301775 Terms a(10) and beyond from _Andrew Howroyd_, Mar 26 2018