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%I #37 Sep 08 2022 08:45:19
%S 1,0,4,12,52,204,820,3276,13108,52428,209716,838860,3355444,13421772,
%T 53687092,214748364,858993460,3435973836,13743895348,54975581388,
%U 219902325556,879609302220,3518437208884,14073748835532
%N Number of closed walks of length n on the complete graph on 5 nodes from a given node.
%H Vincenzo Librandi, <a href="/A109499/b109499.txt">Table of n, a(n) for n = 0..1000</a>
%H Ji Young Choi, <a href="https://www.emis.de/journals/JIS/VOL21/Choi/choi10.html">A Generalization of Collatz Functions and Jacobsthal Numbers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,4).
%F G.f.: (1 - 3*x)/(1 - 3*x - 4*x^2).
%F a(n) = (4^n + 4*(-1)^n)/5.
%F a(n+1) = 4*A015521(n). - _Paul Curtz_, Nov 01 2009
%F a(n) = 3*a(n-1) + 4*a(n-1). - _G. C. Greubel_, Dec 30 2017
%F a(n) = A108020((n - 1) / 2) = 'ccc...c' (n digits) in base 16, for odd n. - _Georg Fischer_, Mar 23 2019
%F E.g.f.: (exp(4*x) + 4*exp(-x))/5. - _G. C. Greubel_, Mar 23 2019
%t CoefficientList[Series[(1-3*x)/(1-3*x-4*x^2), {x,0,30}], x] (* or *) LinearRecurrence[{3,4}, {1,0}, 30] (* _G. C. Greubel_, Dec 30 2017 *)
%o (Magma) [(4^n + 4*(-1)^n)/5: n in [0..30]]; // _Vincenzo Librandi_, Aug 12 2011
%o (PARI) a(n)=(4^n+4*(-1)^n)/5 \\ _Charles R Greathouse IV_, Oct 01 2012
%o (Sage) [(4^n+4*(-1)^n)/5 for n in (0..30)] # _G. C. Greubel_, Mar 23 2019
%o (GAP) a:=[1,0];; for n in [3..30] do a[n]:=3*a[n-1]+4*a[n-2]; od; a; # _G. C. Greubel_, Mar 23 2019
%Y Cf. A108020 (bisection), A109502.
%Y Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521 (forms k=4^n-k). - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%K nonn,easy,walk
%O 0,3
%A _Mitch Harris_, Jun 30 2005
%E Corrected by _Franklin T. Adams-Watters_, Sep 18 2006
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