A199579 Average number of round trips of length n on the Laguerre graph L_4.
1, 4, 28, 232, 2056, 18784, 174112, 1625152, 15220288, 142777600, 1340416768, 12588825088, 118252556800, 1110898849792, 10436554713088, 98050271875072, 921180638875648, 8654518327066624, 81309636020912128
Offset: 0
Examples
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.
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)
Programs
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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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Mathematica
LinearRecurrence[{16, -72, 96, -24}, {1, 4, 28, 232}, 50] (* G. C. Greubel, May 14 2018 *) -
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
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).
Comments