|
%I #18 Jan 11 2024 14:58:20
%S 5,75,500,4375,34375,281250,2265625,18359375,148437500,1201171875,
%T 9716796875,78613281250,635986328125,5145263671875,41625976562500,
%U 336761474609375,2724456787109375,22041320800781250,178318023681640625
%N a(n) = 5^n*Lucas(n), where Lucas = A000204.
%H G. C. Greubel, <a href="/A127212/b127212.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,25).
%F a(n) = Trace of matrix [({5,5},{5,0})^n].
%F a(n) = 5^n * Trace of matrix [({1,1},{1,0})^n].
%F From _Colin Barker_, Sep 02 2013: (Start)
%F a(n) = 5*a(n-1) + 25*a(n-2).
%F G.f.: -5*x*(10*x+1)/(25*x^2+5*x-1). (End)
%t Table[5^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
%t Table[5^n*LucasL[n], {n,1,50}] (* _G. C. Greubel_, Dec 18 2017 *)
%t LinearRecurrence[{5,25},{5,75},20] (* _Harvey P. Dale_, Jan 11 2024 *)
%o (PARI) x='x+O('x^30); Vec(-5*x*(10*x+1)/(25*x^2+5*x-1)) \\ _G. C. Greubel_, Dec 18 2017
%o (Magma) [5^n*Lucas(n): n in [1..30]]; // _G. C. Greubel_, Dec 18 2017
%Y Cf. A000204, A087131, A127210, A127211, A127213, A127214, A127215, A127216.
%K nonn,easy
%O 1,1
%A _Artur Jasinski_, Jan 09 2007
|
