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a(n) = 25*a(n-2) + 48*a(n-3) with a(0) = 0, a(1) = a(2) = 1.
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%I #21 Sep 08 2022 08:45:23

%S 0,1,1,25,73,673,3025,20329,107929,653425,3674017,21516217,123214825,

%T 714258241,4113149041,23770767625,137113121593,791700344593,

%U 4568824885825,26373938451289,152222238686089,878652055801825

%N a(n) = 25*a(n-2) + 48*a(n-3) with a(0) = 0, a(1) = a(2) = 1.

%H G. C. Greubel, <a href="/A114553/b114553.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,25,48).

%F a(n) = w(n)((1)) where w(n) = M*w(n-1), w(0) = {0, 1, 1}, and M = {{0, 1, 0}, {0, 0, 1}, {48, 25, 0}}.

%F From _Colin Barker_, Sep 23 2012: (Start)

%F a(n) = 25*a(n-2) + 48*a(n-3).

%F G.f.: x*(1+x)/((1+3*x)*(1-3*x-16*x^2)). (End)

%F a(n) = (4*i)^(n-1)*(4*i*ChebyshevU(n, -3*I/8) - 5*ChebyshevU(n-1, -3*I/8)) - (-3)^n. - _G. C. Greubel_, Jul 07 2021

%t M = {{0, 1, 0}, {0, 0, 1}, {48, 25, 0}}; w[0] = {0, 1, 1};

%t w[n_]:= w[n]= M.w[n - 1];

%t Table[w[n][[1]], {n, 0, 30}]

%t LinearRecurrence[{0,25,48},{0,1,1},30] (* _Harvey P. Dale_, Mar 26 2013 *)

%o (Magma) I:=[0,1,1]; [n le 3 select I[n] else 25*Self(n-2) + 48*Self(n-3): n in [1..31]]; // _G. C. Greubel_, Jul 07 2021

%o (Sage)

%o def a(n, q): return 0 if (n==0) else 1 if (n<3) else q^2*a(n-2, q) + 2*(q^2-1)*a(n-3, q)

%o [a(n,5) for n in (0..30)] # _G. C. Greubel_, Jul 07 2021

%K nonn,easy

%O 0,4

%A _Roger L. Bagula_, Feb 15 2006

%E New name and edited by _G. C. Greubel_, Jul 07 2021