A201200 Total number of round trips of length n on the closed Laguerre graph Lc_4 divided by 4.
1, 4, 30, 256, 2356, 22384, 215640, 2090176, 20315536, 197702464, 1925042400, 18749072896, 182629124416, 1779030655744, 17330352562560, 168824779580416, 1644626142474496, 16021353180980224, 156074394613317120, 1520422660926324736
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Wolfdieter Lang, Counting walks on Jacobi graphs: an application of orthogonal polynomials.
- Index entries for linear recurrences with constant coefficients, signature (16,-68,64,44).
Programs
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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
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Mathematica
LinearRecurrence[{16,-68,64,44}, {1, 4, 30, 256}, 30] (* G. C. Greubel, May 13 2018 *) -
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
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
Extensions
Typo in formula fixed by Colin Barker, Apr 27 2016
Comments