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%I #18 Sep 08 2022 08:45:27
%S 1,13,50,149,409,1090,2873,7541,19762,51757,135521,354818,928945,
%T 2432029,6367154,16669445,43641193,114254146,299121257,783109637,
%U 2050207666,5367513373,14052332465,36789484034,96316119649,252158874925
%N Expansion of x*(1+9*x+2*x^2)/((1-x)*(1-3*x+x^2)).
%H G. C. Greubel, <a href="/A121990/b121990.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F a(n) = 3*a(n - 1) - a(n - 2) + 12.
%F a(n) = (1/10)*(-120 + (65 - 11*sqrt(5))*((1/2)*(3 - sqrt(5)))^n + ((1/2)*(3 + sqrt(5)))^n*(65 + 11*sqrt(5))).
%F From _R. J. Mathar_, Apr 04 2009: (Start)
%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
%F G.f.: x*(1+9*x+2*x^2)/((1-x)*(1-3*x+x^2)). (End)
%F a(n) = 12*Fibonacci(2*n-1) + Fibonacci(2*n-3) - 12. - _G. C. Greubel_, Nov 21 2019
%p with(combinat); seq(12*fibonacci(2*n-1) +fibonacci(2*n-3) -12, n=1..30); # _G. C. Greubel_, Nov 21 2019
%t LinearRecurrence[{4,-4,1}, {1,13,50}, 30] (* _G. C. Greubel_, Sep 14 2017 *)
%t With[{F=Fibonacci}, Table[12*(F[2*n-1]-1) + F[2*n-3], {n,30}]] (* _G. C. Greubel_, Nov 21 2019 *)
%o (PARI) x='x+O('x^30); Vec(x*(1+9*x+2*x^2)/((1-x)*(x^2-3*x+1))) \\ _G. C. Greubel_, Sep 14 2017
%o (PARI) vector(30, n, 12*fibonacci(2*n-1) +fibonacci(2*n-3) -12) \\ _G. C. Greubel_, Nov 21 2019
%o (Magma) F:= Fibonacci; [12*F(2*n-1) +F(2*n-3) -12: n in [1..30]]; // _G. C. Greubel_, Nov 21 2019
%o (Sage) f=fibonacci; [12*f(2*n-1) + f(2*n-3) -12 for n in (1..30)] # _G. C. Greubel_, Nov 21 2019
%o (GAP) F:=Fibonacci;; List([1..30], n-> 12*F(2*n-1) +F(2*n-3) -12 ); # _G. C. Greubel_, Nov 21 2019
%Y Cf. A000045, A003215, A005891.
%K nonn
%O 1,2
%A _Roger L. Bagula_, Sep 10 2006
%E Edited and new name based on g.f. by _G. C. Greubel_ and _Joerg Arndt_, Sep 14 2017
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