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A198635 Total number of round trips, each of length 2*n on the graph P_5 (o-o-o-o-o).

%I A198635 #51 Aug 10 2025 14:57:42
%S A198635 5,8,20,56,164,488,1460,4376,13124,39368,118100,354296,1062884,
%T A198635 3188648,9565940,28697816,86093444,258280328,774840980,2324522936,
%U A198635 6973568804,20920706408,62762119220,188286357656,564859072964,1694577218888,5083731656660,15251194969976
%N A198635 Total number of round trips, each of length 2*n on the graph P_5 (o-o-o-o-o).
%C A198635 See the array and triangle A198632 for the general case for the graph P_N (there N is n and the length is l = 2*k).
%H A198635 Vincenzo Librandi, <a href="/A198635/b198635.txt">Table of n, a(n) for n = 0..1000</a>
%H A198635 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A198635 a(n) = w(5,2*n), n >= 0, with w(5,l) the total number of closed walks on the graph P_5 (the simple path with 5 points (vertices) and 4 lines (or edges)).
%F A198635 O.g.f. for w(5,l) (with zeros for odd l): y*(d/dy)S(5,y)/S(5,y) with y = 1/x and Chebyshev S-polynomials (coefficients A049310). See also A198632 for a rewritten form.
%F A198635 G.f.: (5-12*x+3*x^2)/(1-4*x+3*x^2). - _Colin Barker_, Jan 02 2012
%F A198635 a(n) = 3*a(n-1) - 4, n > 1. - _Vincenzo Librandi_, Jan 02 2012
%F A198635 a(n) = 2*3^n + 2 for n > 0. - _Andrew Howroyd_, Mar 18 2017
%F A198635 a(n) = 2*A034472(n) for n > 0. - _Andrew Howroyd_, Mar 18 2017
%e A198635 With the graph P_5 as 1-2-3-4-5:
%e A198635 n=0: 5, from the length 0 walks starting at 1,2,3,4 and 5.
%e A198635 n=1: 8, from the walks of length 2, namely 121, 212, 232, 323, 343, 434, 454 and 545.
%t A198635 a[0] = 5; a[n_] := 2*3^n + 2; Array[a, 28, 0] (* _Jean-François Alcover_, Nov 01 2017, after _Andrew Howroyd_ *)
%t A198635 CoefficientList[Series[(5 - 12 x + 3 x^2)/(1 - 4 x + 3 x^2), {x, 0, 27}], x] (* _Michael De Vlieger_, Dec 18 2017 *)
%t A198635 LinearRecurrence[{4,-3},{5,8,20},30] (* _Harvey P. Dale_, Nov 27 2024 *)
%Y A198635 Cf. A005248, A198632, A198633.
%Y A198635 Essentially the same as A115099.
%K A198635 nonn,easy
%O A198635 0,1
%A A198635 _Wolfdieter Lang_, Nov 02 2011