(MAGMAMagma) R<x>:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!( x/(1-(x+x^2)^3) )); // G. C. Greubel, Mar 22 2019

proposed

approved

editing

proposed

(* Gross page 62 voltage group L3 : weights set to one *)

(*Gross page 62 voltage group L3 : weights set to one*) 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}]

(* alternate program *)

proposed

editing

editing

proposed

Edited by G. C. Greubel, Mar 22 2019

Lower level digraph derived from a voltage graph (Gross's covering graph construction) that is generalized Fibonacci Markov in matrix terms to give a 6 X 6 Markov: Characteristic Polynomial: (-1 - x + x^2)(1 + 2 x + 2 x^2 + x^3 + x^4).

Lower level digraph derived from a voltage graph.

Ratio=limit[a(n+1)/a(n),n->Infinity]=Golden Mean This Lower level digraph derived from a voltage graph (Gross's covering graph construction gives ) that is a complex substructure to the generalized Fibonacci Pisot that is not PisotMarkov. Gross's covering graph constructions called voltage graphs are abstractions from lower level graphsIn 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).

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.

limit_{n to Infinity} (a(n+1)/a(n)) = Golden Mean.

G. C. Greubel, <a href="/A115055/b115055.txt">Table of n, a(n) for n = 1..1000</a>

<a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,3,3,1).

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).

From Vladimir Kruchinin, Oct 12 2011: (Start)

G.f.: x/(1-(x+x^2)^3).

G.f.: x/(1-(x+x^2)^3). a(n) =sum( Sum_{k=0..n, } binomial(3*k,n-3*k)). [From Vladimir Kruchinin, Oct 12 2011](End)

a(n) = a(n-3) + 3*a(n-4) + 3*a(n-5) + a(n-6). - G. C. Greubel, Mar 22 2019

LinearRecurrence[{0, 0, 1, 3, 3, 1}, {0, 1, 0, 0, 1, 3}, 50] (* G. C. Greubel, Mar 22 2019 *)

(PARI) my(x='x+O('x^50)); concat([0], Vec(x/(1-(x+x^2)^3))) \\ G. C. Greubel, Mar 22 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!( x/(1-(x+x^2)^3) )); // G. C. Greubel, Mar 22 2019

(Sage) (x/(1-(x+x^2)^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Mar 22 2019

(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

nonn,uned

nonn

approved

editing

_Roger L. Bagula_, Dec 09 2006

reviewed

approved

proposed

reviewed