OFFSET
1,6
COMMENTS
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).
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.
REFERENCES
J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987; see Figure 2.5 p. 62
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1,3,3,1).
FORMULA
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).
a(n) = Sum_{k=0..n} binomial(3*k,n-3*k). (End)
a(n) = a(n-3) + 3*a(n-4) + 3*a(n-5) + a(n-6). - G. C. Greubel, Mar 22 2019
MATHEMATICA
(* 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 *)
LinearRecurrence[{0, 0, 1, 3, 3, 1}, {0, 1, 0, 0, 1, 3}, 50] (* G. C. Greubel, Mar 22 2019 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Dec 09 2006
EXTENSIONS
Edited by G. C. Greubel, Mar 22 2019
STATUS
approved