

A199579


Average number of round trips of length n on the Laguerre graph L_4.


2



1, 4, 28, 232, 2056, 18784, 174112, 1625152, 15220288, 142777600, 1340416768, 12588825088, 118252556800, 1110898849792, 10436554713088, 98050271875072, 921180638875648, 8654518327066624, 81309636020912128
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OFFSET

0,2


COMMENTS

See the general array and triangle for the average number of round trips of length L on (open) Laguerre graphs L_N given in A201198. Here a(n) = w(4,L=n), n>=0, the fourth row in this array. In the corresponding triangle this is the column no. N=4 without leading zeros: a(n) = A201198(n+3,4), n>=0.
For a sketch of this Laguerre graph L_4 see Figure 3 of the W. Lang link. The o.g.f. is also given there.
By definition the number of zero length round trips of length 0 for a vertex is put to 1 in order to count vertices.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..450
Wolfdieter Lang, Counting walks on Jacobi graphs: an application of orthogonal polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,72,96,24)


FORMULA

a(n) = A201198(n+3,4), n>=0.
O.g.f.: (112*x+36*x^224*x^3)/(116*x+72*x^296*x^3+24*x^4).


EXAMPLE

n=0: a(0)=1 because the average number of vertices is 4/4=1.
a(1)= (1+3+5+7)/4 = 4, from the sum of the selfloops of L_4 divided by the number of vertices 4.
The counting for n=2, a(2)= 112/4 = 28, has been given as an example to A201198.


MATHEMATICA

LinearRecurrence[{16, 72, 96, 24}, {1, 4, 28, 232}, 50] (* G. C. Greubel, May 14 2018 *)


PROG

(PARI) x='x+O('x^30); Vec((112*x+36*x^224*x^3)/(116*x+72*x^2 96*x^3 +24*x^4)) \\ G. C. Greubel, May 14 2018
(MAGMA) I:=[1, 4, 28, 232]; [n le 4 select I[n] else 16*Self(n1)  72*Self(n2) + 96*Self(n3) 24*Self(n4): n in [1..30]]; // G. C. Greubel, May 14 2018


CROSSREFS

Cf. A201198, A201199 (closed Laguerre graphs), A201200 (closed L_4 graph).
Sequence in context: A191094 A220876 A152599 * A089023 A035610 A229652
Adjacent sequences: A199576 A199577 A199578 * A199580 A199581 A199582


KEYWORD

nonn,easy,walk


AUTHOR

Wolfdieter Lang, Dec 02 2011


STATUS

approved



