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
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
KEYWORD
nonn,easy,walk
AUTHOR
Wolfdieter Lang, Dec 02 2011
EXTENSIONS
Typo in formula fixed by Colin Barker, Apr 27 2016
STATUS
approved