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A122690
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a(n) = 5*a(n-1) + 4*a(n-2) with a(0)=1, a(1)=4.
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6
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1, 4, 24, 136, 776, 4424, 25224, 143816, 819976, 4675144, 26655624, 151978696, 866515976, 4940494664, 28168537224, 160604664776, 915697472776, 5220906022984, 29767320006024, 169720224122056, 967670400634376
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} 4^k*A122542(n,k).
G.f.: (1-x)/(1-5*x-4*x^2).
Lim_{n -> infinity} a(n+1)/a(n)-> (5+sqrt(41))/2 = 5.701562118716.
a(n) = 2^(n-1)*(2*Fibonacci(n+1, 5/2) - Fibonacci(n, 5/2)) = -(2/I)^(n-1)*( 2*I*ChebyshevU(n, 5*I/4) + ChebyshevU(n-1, 5*I/4)). - G. C. Greubel, Dec 26 2019
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MAPLE
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seq(coeff(series((1-x)/(1-5*x-4*x^2), x, n+1), x, n), n = 0..35); # G. C. Greubel, Dec 26 2019
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MATHEMATICA
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LinearRecurrence[{5, 4}, {1, 4}, 35] (* Harvey P. Dale, Apr 06 2012 *)
Table[2^(n-1)*(2*Fibonacci[n+1, 5/2] - Fibonacci[n, 5/2]), {n, 0, 35}] (* G. C. Greubel, Dec 26 2019 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1-x)/(1-5*x-4*x^2) )); // G. C. Greubel, Dec 26 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x)/(1-5*x-4*x^2) ).list()
(GAP) a:=[1, 4];; for n in [3..35] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019
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CROSSREFS
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KEYWORD
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nonn,easy,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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