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A198633 Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).

%I A198633 #48 Aug 10 2025 14:58:05
%S A198633 3,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A198633 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,
%U A198633 33554432,67108864,134217728,268435456,536870912,1073741824,2147483648
%N A198633 Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).
%C A198633 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 A198633 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%F A198633 a(n) = w(3,2*n), n>=0, with w(3,l) the total number of closed walks on the graph P_3 (the simple path with 3 points (vertices) and 2 lines (or edges)).
%F A198633 O.g.f. for w(3,l) (with zeros for odd l): y*(d/dy)S(3,y)/S(3,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See A198632, also for a rewritten form.
%F A198633 Empirical g.f.: (3-2*x)/(1-2*x). - _Colin Barker_, Jan 02 2012
%F A198633 This g.f. follows from the Chebyshev o.g.f. given above with x -> sqrt(x). Therefore a(0) = 3 and a(n) = 2^(n+1), n >= 1. - _Wolfdieter Lang_, Feb 18 2013.
%e A198633 With the graph P_3 as 1-2-3:
%e A198633 n=0: 3, from the length 0 walks starting at 1, 2 and 3.
%e A198633 n=2: 8, from the walks of length 4, namely 12121, 12321, 21212, 23232, 21232, 23212, 32323 and 32123.
%t A198633 Join[{3},NestList[2#&,4,30]] (* _Harvey P. Dale_, Nov 07 2020 *)
%o A198633 (PARI) a(n)=if(n,2<<n,3) \\ _Charles R Greathouse IV_, Jan 02 2012
%Y A198633 Cf. A198632, 2*A005248, A198635.
%Y A198633 Essentially the same as A000079, A020707, A077552 etc.
%K A198633 nonn,easy
%O A198633 0,1
%A A198633 _Wolfdieter Lang_, Nov 02 2011