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%I #28 Apr 15 2024 13:09:03
%S 2,4,48,256,1792,11264,73728,475136,3080192,19922944,128974848,
%T 834666496,5402263552,34963718144,226291089408,1464583847936,
%U 9478992822272,61349312856064,397061136580608,2569833552019456,16632312393367552,107646586405781504,696703343917006848
%N a(n) = 4^n*Lucas(n), where Lucas = A000032.
%H G. C. Greubel, <a href="/A127211/b127211.txt">Table of n, a(n) for n = 0..1225</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,16).
%F a(n) = Trace of matrix [({4,4},{4,0})^n].
%F a(n) = 4^n * Trace of matrix [({1,1},{1,0})^n].
%F From _Colin Barker_, Sep 02 2013: (Start)
%F a(n) = 4*a(n-1) + 16*a(n-2).
%F G.f.: 2*x*(2*x-1)/(16*x^2+4*x-1). (End)
%F From _Peter Luschny_, Apr 15 2024: (Start)
%F a(n) = 2^n*((1 - sqrt(5))^n + (1 + sqrt(5))^n).
%F a(n) = 4^n*(Fibonacci(n+1) + Fibonacci(n-1)). (End)
%F a(n) = 2^n*A087131(n). - _Michel Marcus_, Apr 15 2024
%p a:= n-> 4^n*(<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
%p seq(a(n), n=0..22); # _Alois P. Heinz_, Apr 15 2024
%t Table[4^n Tr[MatrixPower[{{1, 1}, {1, 0}}, n]], {n, 0, 20}]
%t Table[4^n*LucasL[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 18 2017 *)
%o (PARI) my(x='x + O('x^30)); Vec(-4*x*(8*x+1)/(16*x^2+4*x-1)) \\ _G. C. Greubel_, Dec 18 2017
%o (Magma) [4^n*Lucas(n): n in [0..30]]; // _G. C. Greubel_, Dec 18 2017
%Y Cf. A000032, A000204, A087131, A087131, A127210, A127212, A127213, A127214, A127215, A127216.
%K nonn,easy
%O 0,1
%A _Artur Jasinski_, Jan 09 2007
%E a(0)=2 prepended by _Alois P. Heinz_, Apr 15 2024
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