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A115055 Lower level digraph derived from a voltage graph.

%I A115055 #21 Sep 08 2022 08:45:23
%S A115055 0,1,0,0,1,3,3,2,6,15,21,24,42,86,138,192,305,546,906,1381,2175,3651,
%T A115055 6042,9582,15225,24901,40836,65748,105364,170796,278184,450017,724968,
%U A115055 1172412,1902321,3080367,4975551,8044478,13029534,21096027,34114553
%N A115055 Lower level digraph derived from a voltage graph.
%C A115055 Lower level digraph derived from a voltage graph (Gross's covering graph construction) that is a generalized Fibonacci Markov. In matrix terms gives a 6 X 6 Markov with characteristic Polynomial (-1 - x + x^2)*(1 + 2*x + 2*x^2 + x^3 + x^4).
%C A115055 This digraph construction gives a complex substructure to the Fibonacci Pisot that is not Pisot. Gross's covering graph constructions called voltage graphs are abstractions from lower level graphs.
%C A115055 limit_{n to Infinity} (a(n+1)/a(n)) = Golden Mean.
%D A115055 J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987; see Figure 2.5 p. 62
%H A115055 G. C. Greubel, <a href="/A115055/b115055.txt">Table of n, a(n) for n = 1..1000</a>
%H A115055 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,3,3,1).
%F A115055 Let M be the 6x6 matrix given by: M = {{0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 1, 0}}, then v(n) = M.v(n-1), where a(n) = v(n)(1).
%F A115055 From _Vladimir Kruchinin_, Oct 12 2011: (Start)
%F A115055 G.f.: x/(1-(x+x^2)^3).
%F A115055 a(n) = Sum_{k=0..n} binomial(3*k,n-3*k). (End)
%F A115055 a(n) = a(n-3) + 3*a(n-4) + 3*a(n-5) + a(n-6). - _G. C. Greubel_, Mar 22 2019
%t A115055 (* Gross page 62 voltage group L3 : weights set to one *)
%t A115055 M = {{0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 1, 0}} v[1] = {0, 0, 0, 0, 0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
%t A115055 (* alternate program *)
%t A115055 LinearRecurrence[{0,0,1,3,3,1}, {0,1,0,0,1,3}, 50] (* _G. C. Greubel_, Mar 22 2019 *)
%o A115055 (PARI) my(x='x+O('x^50)); concat([0], Vec(x/(1-(x+x^2)^3))) \\ _G. C. Greubel_, Mar 22 2019
%o A115055 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!( x/(1-(x+x^2)^3) )); // _G. C. Greubel_, Mar 22 2019
%o A115055 (Sage) (x/(1-(x+x^2)^3)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 22 2019
%o A115055 (GAP) a:=[0,1,0,0,1,3];; for n in [7..50] do a[n]:=a[n-3]+3*a[n-4]+ 3*a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Mar 22 2019
%K A115055 nonn
%O A115055 1,6
%A A115055 _Roger L. Bagula_, Dec 09 2006
%E A115055 Edited by _G. C. Greubel_, Mar 22 2019