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
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
KEYWORD
nonn,easy,walk
AUTHOR
Wolfdieter Lang, Dec 02 2011
STATUS
approved