

A115055


Lower level digraph derived from a voltage graph.


2



0, 1, 0, 0, 1, 3, 3, 2, 6, 15, 21, 24, 42, 86, 138, 192, 305, 546, 906, 1381, 2175, 3651, 6042, 9582, 15225, 24901, 40836, 65748, 105364, 170796, 278184, 450017, 724968, 1172412, 1902321, 3080367, 4975551, 8044478, 13029534, 21096027, 34114553
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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(n1), 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,n3*k). (End)
a(n) = a(n3) + 3*a(n4) + 3*a(n5) + a(n6).  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[n3]+3*a[n4]+ 3*a[n5]+a[n6]; od; a; # G. C. Greubel, Mar 22 2019


CROSSREFS

Sequence in context: A200174 A266153 A086636 * A158468 A238278 A200770
Adjacent sequences: A115052 A115053 A115054 * A115056 A115057 A115058


KEYWORD

nonn


AUTHOR

Roger L. Bagula, Dec 09 2006


EXTENSIONS

Edited by G. C. Greubel, Mar 22 2019


STATUS

approved



