A201200 Total number of round trips of length n on the closed Laguerre graph Lc_4 divided by 4.

1, 4, 30, 256, 2356, 22384, 215640, 2090176, 20315536, 197702464, 1925042400, 18749072896, 182629124416, 1779030655744, 17330352562560, 168824779580416, 1644626142474496, 16021353180980224, 156074394613317120, 1520422660926324736

Offset

0,2

Comments

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.

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.

By definition the number of length 0 round trips for a vertex is put to 1 in order to count vertices.

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.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Wolfdieter Lang, Counting walks on Jacobi graphs: an application of orthogonal polynomials.

Index entries for linear recurrences with constant coefficients, signature (16,-68,64,44).

Formula

a(n) = A201199(n+3,4)/4, n>=0.

O.g.f.: (8*x-1)*(2*x^2-4*x+1) / ( (22*x^2-12*x+1)*(2*x^2+4*x-1) ).

From Colin Barker, Apr 27 2016: (Start)

a(n) = 16*a(n-1)-68*a(n-2)+64*a(n-3)+44*a(n-4) for n>3.

a(n) = ((2-sqrt(6))^n+(2+sqrt(6))^n+(6-sqrt(14))^n+(6+sqrt(14))^n)/4.

(End)

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

Mathematica

LinearRecurrence[{16, -68, 64, 44}, {1, 4, 30, 256}, 30] (* G. C. Greubel, May 13 2018 *)

Prog

(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
(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

Crossrefs

Cf. A201199, A201198 (open Laguerre graphs). A199579 (open L_4 graph).

Sequence in context: A086452 A334719 A091527 * A102307 A209441 A371486

Adjacent sequences: A201197 A201198 A201199 * A201201 A201202 A201203

Keyword

nonn,easy,walk

Author

Wolfdieter Lang, Dec 02 2011

Extensions

Typo in formula fixed by Colin Barker, Apr 27 2016

Status

approved