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%I #46 Sep 08 2022 08:45:01
%S 1,8,23,61,160,419,1097,2872,7519,19685,51536,134923,353233,924776,
%T 2421095,6338509,16594432,43444787,113739929,297775000,779585071,
%U 2040980213,5343355568,13989086491,36623903905,95882625224,251023971767,657189290077,1720543898464
%N a(n) = 3*a(n-1) - a(n-2) with a(0) = 1, a(1) = 8.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H Vincenzo Librandi, <a href="/A055273/b055273.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F a(n) = (8*(((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n) - (((3 + sqrt(5))/2)^(n - 1) - ((3 - sqrt(5))/2)^(n - 1)))/sqrt(5).
%F G.f.: (1 + 5*x)/(1 - 3*x + x^2).
%F From _Rigoberto Florez_, Dec 24 2018: (Start)
%F a(n) = Fibonacci(2n+2) + 5*Fibonacci(2n),
%F a(n) = 3*Fibonacci(2n+2) - Fibonacci(2n-3). (End)
%F E.g.f.: (1/5)*exp(3*x/2)*(5*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)). - _Franck Maminirina Ramaharo_, Dec 26 2018
%F a(n) = ChebyshevT(n, 3/2) + (13/2)*ChebyshevU(n-1, 3/2) = ChebyshevU(n, 3/2) + 5*ChebyshevU(n-1, 3/2). - _G. C. Greubel_, Jan 17 2020
%p seq(coeff(series((1+5*x)/(1-3*x+x^2),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Dec 29 2018
%t Table[3Fibonacci[2n+2]-Fibonacci[2n-3], {n,0,20}] (* _Rigoberto Florez_, Dec 24 2018 *)
%t LinearRecurrence[{3, -1}, {1, 8}, 30] (* _Vincenzo Librandi_, Dec 25 2018 *)
%o (Magma) [Fibonacci(2*n+2) + 5*Fibonacci(2*n): n in [0..30]]; // _Vincenzo Librandi_, Dec 25 2018
%o (GAP) a:=[1,8];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Print(a); # _Muniru A Asiru_, Dec 29 2018
%o (PARI) vector(30, n, fibonacci(2*n) + 5*fibonacci(2*n-2) ) \\ _G. C. Greubel_, Jan 17 2020
%o (Sage) [fibonacci(2*n+2) +5*fibonacci(2*n) for n in (0..30)] # _G. C. Greubel_, Jan 17 2020
%Y Cf. A000032, A000045, A055267.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 28 2000
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