

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
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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*x1)*(2*x^24*x+1) / ( (22*x^212*x+1)*(2*x^2+4*x1) ).
From Colin Barker, Apr 27 2016: (Start)
a(n) = 16*a(n1)68*a(n2)+64*a(n3)+44*a(n4) for n>3.
a(n) = ((2sqrt(6))^n+(2+sqrt(6))^n+(6sqrt(14))^n+(6+sqrt(14))^n)/4.
(End)
E.g.f.: (exp((2sqrt(6))*x) + exp((2+sqrt(6))*x) + exp((6sqrt(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((18*x)*(14*x+2*x^2)/((14*x2*x^2)*(112*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(n1)  68*Self(n2) + 64*Self(n3) + 44*Self(n4): 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



