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A122690 a(n) = 5*a(n-1) + 4*a(n-2) with a(0)=1, a(1)=4.

%I A122690 #25 Dec 31 2023 11:29:40
%S A122690 1,4,24,136,776,4424,25224,143816,819976,4675144,26655624,151978696,
%T A122690 866515976,4940494664,28168537224,160604664776,915697472776,
%U A122690 5220906022984,29767320006024,169720224122056,967670400634376
%N A122690 a(n) = 5*a(n-1) + 4*a(n-2) with a(0)=1, a(1)=4.
%H A122690 Harvey P. Dale, <a href="/A122690/b122690.txt">Table of n, a(n) for n = 0..1000</a>
%H A122690 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,4).
%F A122690 a(n) = Sum_{k=0..n} 4^k*A122542(n,k).
%F A122690 G.f.: (1-x)/(1-5*x-4*x^2).
%F A122690 Lim_{n -> infinity} a(n+1)/a(n)-> (5+sqrt(41))/2 = 5.701562118716.
%F A122690 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
%p A122690 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
%t A122690 LinearRecurrence[{5,4},{1,4},35] (* _Harvey P. Dale_, Apr 06 2012 *)
%t A122690 Table[2^(n-1)*(2*Fibonacci[n+1, 5/2] - Fibonacci[n, 5/2]), {n,0,35}] (* _G. C. Greubel_, Dec 26 2019 *)
%o A122690 (PARI) Vec((1-x)/(1-5*x-4*x^2)+O(x^35)) \\ _Charles R Greathouse IV_, Jan 17 2012
%o A122690 (Magma) R<x>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1-x)/(1-5*x-4*x^2) )); // _G. C. Greubel_, Dec 26 2019
%o A122690 (Sage)
%o A122690 def A122690_list(prec):
%o A122690     P.<x> = PowerSeriesRing(ZZ, prec)
%o A122690     return P( (1-x)/(1-5*x-4*x^2) ).list()
%o A122690 A122690_list(35) # _G. C. Greubel_, Dec 26 2019
%o A122690 (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
%Y A122690 Cf. A015537.
%K A122690 nonn,easy,less
%O A122690 0,2
%A A122690 _Philippe Deléham_, Sep 22 2006
%E A122690 Corrected by _T. D. Noe_, Nov 07 2006