Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #25 Sep 08 2022 08:46:01
%S 1,4,30,256,2356,22384,215640,2090176,20315536,197702464,1925042400,
%T 18749072896,182629124416,1779030655744,17330352562560,
%U 168824779580416,1644626142474496,16021353180980224,156074394613317120,1520422660926324736
%N Total number of round trips of length n on the closed Laguerre graph Lc_4 divided by 4.
%C For the general array and triangle for the total number of round trips of length L on closed Laguerre graphs Lc_N see A201199. Here a(n)=w(4,L=n)/4, n>=0, the fourth row of this array divided by 4. In the corresponding triangle a(n) = A201199(n+3,4)/4, n>=0.
%C For a sketch of the closed Laguerre graph Lc_4 see Figure 4 of the given W. Lang link. The o.g.f. is also found there.
%C By definition the number of length 0 round trips for a vertex is put to 1 in order to count vertices.
%C The average number of round trips of length n on a closed Laguerre graph Lc_N is in general a fraction. Therefore A201199 tabulates the total number of round trips.
%H Colin Barker, <a href="/A201200/b201200.txt">Table of n, a(n) for n = 0..1000</a>
%H Wolfdieter Lang, <a href="/A201198/a201198_1.pdf">Counting walks on Jacobi graphs: an application of orthogonal polynomials.</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (16,-68,64,44).
%F a(n) = A201199(n+3,4)/4, n>=0.
%F O.g.f.: (8*x-1)*(2*x^2-4*x+1) / ( (22*x^2-12*x+1)*(2*x^2+4*x-1) ).
%F From _Colin Barker_, Apr 27 2016: (Start)
%F a(n) = 16*a(n-1)-68*a(n-2)+64*a(n-3)+44*a(n-4) for n>3.
%F a(n) = ((2-sqrt(6))^n+(2+sqrt(6))^n+(6-sqrt(14))^n+(6+sqrt(14))^n)/4.
%F (End)
%F E.g.f.: (exp((2-sqrt(6))*x) + exp((2+sqrt(6))*x) + exp((6-sqrt(14))*x) + exp((6+sqrt(14))*x))/4. - _Ilya Gutkovskiy_, Apr 27 2016
%t LinearRecurrence[{16,-68,64,44}, {1, 4, 30, 256}, 30] (* _G. C. Greubel_, May 13 2018 *)
%o (PARI) Vec((1-8*x)*(1-4*x+2*x^2)/((1-4*x-2*x^2)*(1-12*x+22*x^2)) + O(x^50)) \\ _Colin Barker_, Apr 27 2016
%o (Magma) I:=[1, 4, 30, 256]; [n le 4 select I[n] else 16*Self(n-1) - 68*Self(n-2) + 64*Self(n-3) + 44*Self(n-4): n in [1..30]]; // _G. C. Greubel_, May 13 2018
%Y Cf. A201199, A201198 (open Laguerre graphs). A199579 (open L_4 graph).
%K nonn,easy,walk
%O 0,2
%A _Wolfdieter Lang_, Dec 02 2011
%E Typo in formula fixed by _Colin Barker_, Apr 27 2016