%I #58 Oct 03 2024 06:09:54

%S 1,20,336,5440,87296,1397760,22368256,357908480,5726601216,

%T 91625881600,1466015154176,23456246661120,375299963355136,

%U 6004799480791040,96076791961092096,1537228672451215360,24595658763514413056,393530540233410478080,6296488643803287126016

%N a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 20.

%C Partial sums of A166965.

%C First differences of A006105. - _Klaus Purath_, Oct 15 2020

%H Seiichi Manyama, <a href="/A166984/b166984.txt">Table of n, a(n) for n = 0..830</a> (terms 0..200 from Vincenzo Librandi)

%H E. Saltürk and I. Siap, <a href="http://www.ejpam.com/index.php/ejpam/article/view/1609">Generalized Gaussian Numbers Related to Linear Codes over Galois Rings</a>, European Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 2012, 250-259; ISSN 1307-5543. - From _N. J. A. Sloane_, Oct 23 2012

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-64).

%F a(n) = (4*16^n - 4^n)/3.

%F G.f.: 1/((1-4*x)*(1-16*x)).

%F Limit_{n -> oo} a(n)/a(n-1) = 16.

%F a(n) = A115490(n+1)/3.

%F Sum_{n>=0} a(n) x^(2*n+4)/(2*n+4)! = ( sinh(x) )^4/4!. - _Robert A. Russell_, Apr 03 2013

%F From _Klaus Purath_, Oct 15 2020: (Start)

%F a(n) = A002450(n+1)*(A002450(n+2) - A002450(n))/5.

%F a(n) = (A083584(n+1)^2 - A083584(n)^2)/80. (End)

%F a(n) = (A079598(n) - A000302(n))/24. - _César Aguilera_, Jun 21 2022

%F a(n) = 16*a(n-1) + 4^n with a(0) = 1. - _Nadia Lafreniere_, Aug 08 2022

%F E.g.f.: (4/3)*exp(10*x)*sinh(6*x + log(2)). - _G. C. Greubel_, Oct 02 2024

%t LinearRecurrence[{20,-64},{1,20},30] (* _Harvey P. Dale_, Jul 04 2012 *)

%o (Magma) [n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];

%o (PARI) a(n) = (4*16^n - 4^n)/3 \\ _Charles R Greathouse IV_, Jun 21 2022

%o (SageMath)

%o A166984=BinaryRecurrenceSequence(20,-64,1,20)

%o [A166984(n) for n in range(31)] # _G. C. Greubel_, Oct 02 2024

%Y Cf. A000302, A002450, A002452, A002453, A006105, A079598, A083584.

%Y Cf. A115490, A166914, A166917, A166927, A166965, A307695.

%K nonn,easy

%O 0,2

%A _Klaus Brockhaus_, Oct 26 2009