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A201200 Total number of round trips of length n on the closed Laguerre graph Lc_4 divided by 4. 2
1, 4, 30, 256, 2356, 22384, 215640, 2090176, 20315536, 197702464, 1925042400, 18749072896, 182629124416, 1779030655744, 17330352562560, 168824779580416, 1644626142474496, 16021353180980224, 156074394613317120, 1520422660926324736 (list; graph; refs; listen; history; text; internal format)
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: A038225 A086452 A091527 * A102307 A209441 A330801

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

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Last modified April 4 21:43 EDT 2020. Contains 333238 sequences. (Running on oeis4.)