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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 (list; graph; refs; listen; history; text; internal format)
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.: (1-12*x+36*x^2-24*x^3)/(1-16*x+72*x^2-96*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 self-loops 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((1-12*x+36*x^2-24*x^3)/(1-16*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(n-1) - 72*Self(n-2) + 96*Self(n-3) -24*Self(n-4): 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

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Last modified July 5 16:07 EDT 2020. Contains 335473 sequences. (Running on oeis4.)