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A199579
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Average number of round trips of length n on the Laguerre graph L_4.
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2
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1, 4, 28, 232, 2056, 18784, 174112, 1625152, 15220288, 142777600, 1340416768, 12588825088, 118252556800, 1110898849792, 10436554713088, 98050271875072, 921180638875648, 8654518327066624, 81309636020912128
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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O.g.f.: (1-12*x+36*x^2-24*x^3)/(1-16*x+72*x^2-96*x^3+24*x^4).
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EXAMPLE
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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.
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MATHEMATICA
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LinearRecurrence[{16, -72, 96, -24}, {1, 4, 28, 232}, 50] (* G. C. Greubel, May 14 2018 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,walk
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AUTHOR
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STATUS
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approved
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